Numerical methods for the computation of the confluent and Gauss hypergeometric functions

The two most commonly used hypergeometric functions are the confluent hypergeometric function and the Gauss hypergeometric function. We review the available techniques for accurate, fast, and reliable computation of these two hypergeometric functions in different parameter and variable regimes. The...

Full description

Saved in:
Bibliographic Details
Published in:Numerical algorithms Vol. 74; no. 3; pp. 821 - 866
Main Authors: Pearson, John W., Olver, Sheehan, Porter, Mason A.
Format: Journal Article
Language:English
Published: New York Springer US 01.03.2017
Springer Nature B.V
Subjects:
Algebra
Algorithms
Applied mathematics
Asymptotic methods
Asymptotic series
Computation
Computer Science
Differential equations
Hypergeometric functions
Numeric Computing
Numerical Analysis
Numerical methods
Original Paper
Parameters
Quadratures
Recommender systems
Theory of Computation
Confluent hypergeometric function
33C15
Gauss hypergeometric function
Secondary: 41A58
41A60
Primary: 33C05
Computation of special functions
ISSN:1017-1398, 1572-9265
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Abstract The two most commonly used hypergeometric functions are the confluent hypergeometric function and the Gauss hypergeometric function. We review the available techniques for accurate, fast, and reliable computation of these two hypergeometric functions in different parameter and variable regimes. The methods that we investigate include Taylor and asymptotic series computations, Gauss–Jacobi quadrature, numerical solution of differential equations, recurrence relations, and others. We discuss the results of numerical experiments used to determine the best methods, in practice, for each parameter and variable regime considered. We provide “roadmaps” with our recommendation for which methods should be used in each situation.
AbstractList The two most commonly used hypergeometric functions are the confluent hypergeometric function and the Gauss hypergeometric function. We review the available techniques for accurate, fast, and reliable computation of these two hypergeometric functions in different parameter and variable regimes. The methods that we investigate include Taylor and asymptotic series computations, Gauss–Jacobi quadrature, numerical solution of differential equations, recurrence relations, and others. We discuss the results of numerical experiments used to determine the best methods, in practice, for each parameter and variable regime considered. We provide “roadmaps” with our recommendation for which methods should be used in each situation.
Author Porter, Mason A.
Pearson, John W.
Olver, Sheehan
Author_xml – sequence: 1
  givenname: John W.
  surname: Pearson
  fullname: Pearson, John W.
  email: j.w.pearson@kent.ac.uk
  organization: School of Mathematics, Statistics and Actuarial Science, University of Kent
– sequence: 2
  givenname: Sheehan
  surname: Olver
  fullname: Olver, Sheehan
  organization: School of Mathematics and Statistics, The University of Sydney
– sequence: 3
  givenname: Mason A.
  surname: Porter
  fullname: Porter, Mason A.
  organization: Oxford Centre for Industrial and Applied Mathematics, Mathematical Institute, University of Oxford
BookMark eNp9kE9LAzEQxYNUsK1-AG8Bz6uT7J_sHqVoFUQvevAUsruTdstuUpPsod_elC0Igh5CwuT93sy8BZkZa5CQawa3DEDcecZA5AmwIh6RJnBG5iwXPKl4kc_iO1YTllblBVl4vwOIFBdz8vk6Dui6RvV0wLC1rafaOhq2SBs77MegQmcNtfpUMrof0QSqTEvXavSebg97dBu0EY8-VI-mOSL-kpxr1Xu8Ot1L8vH48L56Sl7e1s-r-5ekyXgekjRtU-SVbrJctaDbVpeZVoIrKBjWyDXWdfwroC4Eirau6oyXCDyvSy3iuumS3Ey-e2e_RvRB7uzoTGwpecXKAoqcpVElJlXjrPcOtWy6abfgVNdLBvKYo5xylNFYHnOUEEn2i9y7blDu8C_DJ8ZHrdmg-5npb-gbJ3-Ilg
CitedBy_id crossref_primary_10_3390_math13152354
crossref_primary_10_1016_j_camwa_2024_06_001
crossref_primary_10_1016_j_jcp_2017_11_016
crossref_primary_10_3390_e24050600
crossref_primary_10_1111_sjos_12447
crossref_primary_10_1007_s00477_021_02063_4
crossref_primary_10_1214_19_BA1176
crossref_primary_10_3389_fenrg_2021_692627
crossref_primary_10_1109_TPWRD_2022_3199999
crossref_primary_10_1007_s11075_024_01808_w
crossref_primary_10_1007_s00365_022_09606_0
crossref_primary_10_1007_s11075_019_00741_7
crossref_primary_10_1371_journal_pone_0325059
crossref_primary_10_1109_TVT_2021_3058262
crossref_primary_10_1088_1751_8121_ab749b
crossref_primary_10_2478_jos_2018_0041
crossref_primary_10_1016_j_jmps_2023_105430
crossref_primary_10_1080_00949655_2023_2232913
crossref_primary_10_5194_gchron_7_265_2025
crossref_primary_10_1016_j_cpc_2023_108990
crossref_primary_10_1016_j_amc_2023_127901
crossref_primary_10_1088_1402_4896_ac1027
crossref_primary_10_1088_1478_3975_aba50f
crossref_primary_10_1007_s10208_020_09455_y
crossref_primary_10_1051_0004_6361_201937209
crossref_primary_10_1088_1751_8121_ace6e2
crossref_primary_10_1016_j_amc_2022_127188
crossref_primary_10_1109_JSTSP_2022_3222680
crossref_primary_10_1007_s00211_021_01193_9
crossref_primary_10_1007_s00362_021_01262_z
crossref_primary_10_1007_s11134_024_09906_1
crossref_primary_10_1016_j_jmp_2025_102904
crossref_primary_10_1016_j_powtec_2021_07_081
crossref_primary_10_1090_mcom_3740
crossref_primary_10_1080_00949655_2021_2023873
crossref_primary_10_1002_apxr_202300136
crossref_primary_10_1093_mnras_staf922
crossref_primary_10_1093_mnras_sty2784
crossref_primary_10_1007_s00477_021_02162_2
crossref_primary_10_1111_risa_13551
crossref_primary_10_1109_TAES_2019_2949395
Cites_doi 10.1145/355853.355863
10.1090/S0025-5718-07-01934-5
10.1016/j.optcom.2007.12.047
10.1016/j.cam.2005.01.041
10.1103/PhysRev.163.147
10.1137/06067016X
10.1023/B:NUMA.0000040063.91709.58
10.1017/S0308210500012130
10.1142/S0219530503000028
10.1007/s00211-008-0175-5
10.1063/1.871296
10.1109/TCOMM.2012.081512.110530
10.1093/imanum/drq012
10.1016/j.cpc.2013.03.016
10.1137/0702017
10.1007/BF02309560
10.1016/j.jnnfm.2011.05.006
10.1534/genetics.113.152991
10.1112/plms/s2-17.1.116
10.1137/0521013
10.1016/0045-7930(89)90004-2
10.1016/j.cpc.2007.11.007
10.1137/0731050
10.1137/0518066
10.1137/120889873
10.1137/050625837
10.1006/jcph.1997.5794
10.1137/1009002
10.1145/131766.131774
10.1137/0520052
10.1016/j.nuclphysb.2011.09.015
10.1145/992200.992204
10.1137/050646342
10.1137/0524046
10.1145/569147.569150
10.1103/PhysRevE.63.066117
10.1002/mma.208
10.1007/BF01396306
10.1137/0522095
10.1046/j.1365-8711.2002.05557.x
10.1364/OE.22.032353
10.1016/S0377-0427(02)00627-1
10.1093/imamat/49.3.203
10.1016/j.cam.2009.02.072
10.1090/S0025-5718-07-01945-X
10.1137/120872553
10.1016/S0377-0427(01)00383-1
10.1142/S021953050300003X
10.1007/s10444-012-9283-y
10.1016/0377-0427(92)90129-L
10.1016/j.cpc.2014.10.002
10.1093/imamat/22.2.215
10.1103/PhysRev.35.1303
10.1090/S0025-5718-07-01918-7
10.1090/S0025-5718-69-99647-1
10.1137/0519088
10.1007/s002110100285
10.1016/0010-4655(92)90071-6
10.1016/0010-4655(80)90022-3
10.1142/S0129183193000665
10.1017/S0962492906330012
10.1142/S0219530514500389
10.1016/0010-4655(84)90146-2
10.1016/S0377-0427(00)00347-2
10.1016/j.cam.2005.11.027
10.6028/jres.071B.018
10.1016/S0377-0427(99)00226-5
10.1142/S0219530510001588
10.1016/j.camwa.2008.04.018
10.1137/1.9780898717822
10.1063/1.2939395
10.1090/psapm/048/1314844
10.1145/1145768.1145782
10.1007/978-1-4757-5443-8
10.31855/07868f86-19f
10.1145/236869.237088
10.1017/CBO9781107325937
10.1016/j.amc.2015.01.117
10.4310/MAA.1995.v2.n2.a4
10.4310/MAA.1999.v6.n2.a8
10.1145/2049662.2049668
10.4310/MAA.1999.v6.n3.a2
10.1016/B978-0-12-459950-5.50006-7
10.1002/9781118032572
10.1016/S0096-3003(02)00930-X
ContentType Journal Article
Copyright Springer Science+Business Media New York 2016
Springer Science+Business Media New York 2016.
Copyright_xml – notice: Springer Science+Business Media New York 2016
– notice: Springer Science+Business Media New York 2016.
DBID AAYXX
CITATION
8FE
8FG
ABJCF
AFKRA
ARAPS
AZQEC
BENPR
BGLVJ
CCPQU
DWQXO
GNUQQ
HCIFZ
JQ2
K7-
L6V
M7S
P62
PHGZM
PHGZT
PKEHL
PQEST
PQGLB
PQQKQ
PQUKI
PRINS
PTHSS
DOI 10.1007/s11075-016-0173-0
DatabaseName CrossRef
ProQuest SciTech Collection
ProQuest Technology Collection
Materials Science & Engineering Collection
ProQuest Central UK/Ireland
Advanced Technologies & Computer Science Collection
ProQuest Central Essentials
ProQuest Central Database Suite (ProQuest)
ProQuest Technology Collection
ProQuest One Community College
ProQuest Central Korea
ProQuest Central Student
SciTech Premium Collection
ProQuest Computer Science Collection
Computer Science Database
ProQuest Engineering Collection
Engineering Database
ProQuest Advanced Technologies & Aerospace Collection
ProQuest Central Premium
ProQuest One Academic (New)
ProQuest One Academic Middle East (New)
ProQuest One Academic Eastern Edition (DO NOT USE)
ProQuest One Applied & Life Sciences
ProQuest One Academic (retired)
ProQuest One Academic UKI Edition
ProQuest Central China
Engineering Collection
DatabaseTitle CrossRef
Computer Science Database
ProQuest Central Student
Technology Collection
ProQuest One Academic Middle East (New)
ProQuest Advanced Technologies & Aerospace Collection
ProQuest Central Essentials
ProQuest Computer Science Collection
SciTech Premium Collection
ProQuest One Community College
ProQuest Central China
ProQuest Central
ProQuest One Applied & Life Sciences
ProQuest Engineering Collection
ProQuest Central Korea
ProQuest Central (New)
Engineering Collection
Advanced Technologies & Aerospace Collection
Engineering Database
ProQuest One Academic Eastern Edition
ProQuest Technology Collection
ProQuest SciTech Collection
ProQuest One Academic UKI Edition
Materials Science & Engineering Collection
ProQuest One Academic
ProQuest One Academic (New)
DatabaseTitleList Computer Science Database
Database_xml – sequence: 1
  dbid: BENPR
  name: ProQuest Central
  url: https://www.proquest.com/central
  sourceTypes: Aggregation Database
DeliveryMethod fulltext_linktorsrc
Discipline Applied Sciences
Mathematics
Computer Science
EISSN 1572-9265
EndPage 866
ExternalDocumentID 10_1007_s11075_016_0173_0
GroupedDBID -4Z
-59
-5G
-BR
-EM
-Y2
-~C
.86
.DC
.VR
06D
0R~
0VY
123
1N0
1SB
2.D
203
29N
2J2
2JN
2JY
2KG
2KM
2LR
2P1
2VQ
2~H
30V
4.4
406
408
409
40D
40E
5QI
5VS
67Z
6NX
8TC
8UJ
95-
95.
95~
96X
AAAVM
AABHQ
AACDK
AAHNG
AAIAL
AAJBT
AAJKR
AANZL
AAOBN
AARHV
AARTL
AASML
AATNV
AATVU
AAUYE
AAWCG
AAYIU
AAYQN
AAYTO
AAYZH
ABAKF
ABBBX
ABBXA
ABDZT
ABECU
ABFTD
ABFTV
ABHLI
ABHQN
ABJCF
ABJNI
ABJOX
ABKCH
ABKTR
ABMNI
ABMQK
ABNWP
ABQBU
ABQSL
ABSXP
ABTAH
ABTEG
ABTHY
ABTKH
ABTMW
ABULA
ABWNU
ABXPI
ACAOD
ACBXY
ACDTI
ACGFS
ACHSB
ACHXU
ACIWK
ACKNC
ACMDZ
ACMLO
ACOKC
ACOMO
ACPIV
ACSNA
ACZOJ
ADHHG
ADHIR
ADIMF
ADINQ
ADKNI
ADKPE
ADRFC
ADTPH
ADURQ
ADYFF
ADZKW
AEBTG
AEFIE
AEFQL
AEGAL
AEGNC
AEJHL
AEJRE
AEKMD
AEMSY
AENEX
AEOHA
AEPYU
AESKC
AETLH
AEVLU
AEXYK
AFBBN
AFEXP
AFGCZ
AFKRA
AFLOW
AFQWF
AFWTZ
AFZKB
AGAYW
AGDGC
AGGDS
AGJBK
AGMZJ
AGQEE
AGQMX
AGRTI
AGWIL
AGWZB
AGYKE
AHAVH
AHBYD
AHKAY
AHSBF
AHYZX
AIAKS
AIGIU
AIIXL
AILAN
AITGF
AJBLW
AJRNO
AJZVZ
ALMA_UNASSIGNED_HOLDINGS
ALWAN
AMKLP
AMXSW
AMYLF
AMYQR
AOCGG
ARAPS
ARMRJ
ASPBG
AVWKF
AXYYD
AYJHY
AZFZN
B-.
BA0
BBWZM
BDATZ
BENPR
BGLVJ
BGNMA
BSONS
CAG
CCPQU
COF
CS3
CSCUP
DDRTE
DL5
DNIVK
DPUIP
DU5
EBLON
EBS
EIOEI
EJD
ESBYG
FEDTE
FERAY
FFXSO
FIGPU
FINBP
FNLPD
FRRFC
FSGXE
FWDCC
GGCAI
GGRSB
GJIRD
GNWQR
GQ6
GQ7
GQ8
GXS
H13
HCIFZ
HF~
HG5
HG6
HMJXF
HQYDN
HRMNR
HVGLF
HZ~
I09
IHE
IJ-
IKXTQ
ITM
IWAJR
IXC
IZIGR
IZQ
I~X
I~Z
J-C
J0Z
JBSCW
JCJTX
JZLTJ
K7-
KDC
KOV
KOW
LAK
LLZTM
M4Y
M7S
MA-
N2Q
N9A
NB0
NDZJH
NPVJJ
NQJWS
NU0
O9-
O93
O9G
O9I
O9J
OAM
OVD
P19
P2P
P9O
PF0
PT4
PT5
PTHSS
QOK
QOS
R4E
R89
R9I
RHV
RNI
RNS
ROL
RPX
RSV
RZC
RZE
RZK
S16
S1Z
S26
S27
S28
S3B
SAP
SCJ
SCLPG
SCO
SDH
SDM
SHX
SISQX
SJYHP
SNE
SNPRN
SNX
SOHCF
SOJ
SPISZ
SRMVM
SSLCW
STPWE
SZN
T13
T16
TEORI
TSG
TSK
TSV
TUC
U2A
UG4
UOJIU
UTJUX
UZXMN
VC2
VFIZW
VOH
W23
W48
WK8
YLTOR
Z45
Z7R
Z7X
Z7Z
Z81
Z83
Z88
Z8M
Z8R
Z8T
Z8W
Z92
ZMTXR
ZY4
~EX
AAPKM
AAYXX
ABBRH
ABDBE
ABFSG
ABRTQ
ACSTC
ADHKG
AEZWR
AFDZB
AFFHD
AFHIU
AFOHR
AGQPQ
AHPBZ
AHWEU
AIXLP
ATHPR
AYFIA
CITATION
PHGZM
PHGZT
PQGLB
8FE
8FG
AZQEC
DWQXO
GNUQQ
JQ2
L6V
P62
PKEHL
PQEST
PQQKQ
PQUKI
PRINS
ID FETCH-LOGICAL-c425t-33d3e29fc45ad0fddf84fa72a061ebe2febbc4560b67e7db9b428e025b8f70163
IEDL.DBID RSV
ISICitedReferencesCount 48
ISICitedReferencesURI http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=000395033800009&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
ISSN 1017-1398
IngestDate Wed Nov 05 02:21:37 EST 2025
Sat Nov 29 07:51:03 EST 2025
Tue Nov 18 22:01:15 EST 2025
Fri Feb 21 02:32:28 EST 2025
IsDoiOpenAccess false
IsOpenAccess true
IsPeerReviewed true
IsScholarly true
Issue 3
Keywords Confluent hypergeometric function
33C15
Gauss hypergeometric function
Secondary: 41A58
41A60
Primary: 33C05
Computation of special functions
Language English
LinkModel DirectLink
MergedId FETCHMERGED-LOGICAL-c425t-33d3e29fc45ad0fddf84fa72a061ebe2febbc4560b67e7db9b428e025b8f70163
Notes ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
OpenAccessLink https://www.research.ed.ac.uk/en/publications/f873c8ce-e3fb-4fa6-943b-8ef95462541f
PQID 2918606513
PQPubID 2043837
PageCount 46
ParticipantIDs proquest_journals_2918606513
crossref_citationtrail_10_1007_s11075_016_0173_0
crossref_primary_10_1007_s11075_016_0173_0
springer_journals_10_1007_s11075_016_0173_0
PublicationCentury 2000
PublicationDate 2017-03-01
PublicationDateYYYYMMDD 2017-03-01
PublicationDate_xml – month: 03
  year: 2017
  text: 2017-03-01
  day: 01
PublicationDecade 2010
PublicationPlace New York
PublicationPlace_xml – name: New York
PublicationTitle Numerical algorithms
PublicationTitleAbbrev Numer Algor
PublicationYear 2017
Publisher Springer US
Springer Nature B.V
Publisher_xml – name: Springer US
– name: Springer Nature B.V
References Pastor-Satorras, Vespignani (CR79) 2001; 63
Trefethen, Weideman, Schmelzer (CR101) 2006; 46
Watson (CR104) 1918; 2
Temme (CR92) 2015
CR37
Olver (CR75) 1967; 71
CR35
Hsu (CR41) 1993; 4
Badralexe, Marksteiner, Oh, Freeman (CR7) 1992; 71
Bühring (CR12) 1988; 19
CR34
Gil, Segura (CR29) 2002; 28
Olde Daalhuis, Olver (CR72) 1995; 2
Olde Daalhuis (CR69) 2003; 1
Segura, Temme (CR88) 2008; 111
Mathar (CR60) 2004; 36
Bell, Scott (CR8) 1980; 20
Gavrila (CR27) 1967; 163
Gautschi (CR26) 2002; 139
Golub, Welsch (CR38) 1969; 23
Olver (CR77) 1991; 22
CR40
Deaño, Temme (CR15) 2009; 35
López, Temme (CR52) 2013; 39
Kniehl, Tarasov (CR47) 2012; 854
Olver (CR76) 1965; 2
Mace, Hellberg (CR59) 1995; 2
Temme (CR98) 1978; 22
Gautschi (CR25) 1979; 5
Chiocchia, Gabutti (CR13) 1989; 17
CR58
CR57
CR56
Glaser, Liu, Rokhlin (CR36) 2007; 29
CR55
CR53
Vitalis, Gautier, Dawson, Beaumont (CR102) 2014; 196
Nardin, Perger, Bhalla (CR64) 1992; 18
López, Pagola (CR51) 2010; 233
Weideman (CR105) 2006; 44
Gil, Segura, Temme (CR28) 2006; 190
Abad, Sesma (CR2) 1995; 5
Deaño, Segura (CR14) 2007; 76
CR66
Korotkova (CR48) 2008; 281
CR62
Spouge (CR89) 1994; 31
Temme (CR93) 2003; 153
Lanczos (CR49) 1964; 1
Olver (CR74) 1993; 24
Temme (CR95) 2007; 16
Olde Daalhuis (CR70) 2003; 1
Ferreira, López, Sinusía (CR22) 2006; 197
Luke (CR54) 1977; 11
Dunster (CR17) 1989; 20
CR115
CR116
Schmelzer, Trefethen (CR86) 2007; 45
CR113
CR114
CR73
CR111
CR112
Temme (CR96) 1983; 41
CR110
Jones (CR44) 2001; 24
Weideman, Trefethen (CR106) 2007; 76
Hale, Townsend (CR39) 2013; 35
Dunster (CR16) 1999; 6
CR3
CR6
CR5
Forrey (CR23) 1997; 137
CR9
Wang, Duan, Li, Luan (CR103) 2015; 258
Olde Daalhuis (CR71) 2010; 8
Temme (CR99) 1990; 21
CR87
Nardin, Perger, Bhalla (CR65) 1992; 39
CR85
Temme (CR94) 2000; 121
Khwaja, Olde Daalhuis (CR46) 2014; 12
Olver (CR78) 1980; 86
CR83
CR82
CR80
Torrieri, Valenti (CR100) 2012; 60
Gil, Segura, Temme (CR31) 2012; 34
Huang, Liu (CR42) 2013; 184
Gautschi (CR24) 1967; 9
Rakha, El-Sedy (CR84) 2004; 148
Efimov (CR19) 2014; 22
CR10
CR97
Ibrahim, Rakha (CR43) 2008; 56
Gil, Segura, Temme (CR32) 2011; 31
López (CR50) 1999; 6
CR91
Noble, Thompson (CR67) 1984; 33
CR90
Pierro, Pinto, Spallicci di Filottrano (CR81) 2002; 334
Allasia, Besenghi (CR4) 1987; 39
Gil, Segura (CR30) 2004; 30
Muller (CR63) 2001; 90
Bühring (CR11) 1987; 18
Zhao, Yang (CR109) 2011; 166
Eckart (CR18) 1930; 35
Eremenko, Upadhyay, Thompson, Elster, Nunes, Arbanas, Escher, Hlophe (CR21) 2015; 187
Gil, Segura, Temme (CR33) 2007; 76
Kalla (CR45) 1992; 45
Michel, Stoitsov (CR61) 2008; 178
CR20
Olde Daalhuis (CR68) 1992; 49
Abad, Sesma (CR1) 1999; 101
CR108
CR107
R Badralexe (173_CR7) 1992; 71
AB Olde Daalhuis (173_CR69) 2003; 1
173_CR110
173_CR91
173_CR90
TM Dunster (173_CR17) 1989; 20
V Eremenko (173_CR21) 2015; 187
RJ Mathar (173_CR60) 2004; 36
173_CR97
173_CR87
W Gautschi (173_CR26) 2002; 139
TM Dunster (173_CR16) 1999; 6
173_CR113
173_CR114
A Efimov (173_CR19) 2014; 22
173_CR111
173_CR112
A Deaño (173_CR15) 2009; 35
R Vitalis (173_CR102) 2014; 196
173_CR115
173_CR116
FWJ Olver (173_CR75) 1967; 71
H-Z Huang (173_CR42) 2013; 184
A Gil (173_CR32) 2011; 31
W Bühring (173_CR12) 1988; 19
173_CR80
173_CR82
A Gil (173_CR29) 2002; 28
CJ Noble (173_CR67) 1984; 33
JAC Weideman (173_CR105) 2006; 44
173_CR83
173_CR85
W Gautschi (173_CR25) 1979; 5
A Gil (173_CR30) 2004; 30
GN Watson (173_CR104) 1918; 2
A Deaño (173_CR14) 2007; 76
R Pastor-Satorras (173_CR79) 2001; 63
M Nardin (173_CR65) 1992; 39
173_CR73
MA Rakha (173_CR84) 2004; 148
NM Temme (173_CR98) 1978; 22
O Korotkova (173_CR48) 2008; 281
173_CR66
KE Muller (173_CR63) 2001; 90
GH Golub (173_CR38) 1969; 23
JAC Weideman (173_CR106) 2007; 76
G Allasia (173_CR4) 1987; 39
A Gil (173_CR31) 2012; 34
NM Temme (173_CR99) 1990; 21
NM Temme (173_CR93) 2003; 153
M Nardin (173_CR64) 1992; 18
SL Kalla (173_CR45) 1992; 45
AB Olde Daalhuis (173_CR68) 1992; 49
LN Trefethen (173_CR101) 2006; 46
173_CR62
RL Mace (173_CR59) 1995; 2
J Abad (173_CR1) 1999; 101
F Khwaja (173_CR46) 2014; 12
173_CR55
173_CR57
173_CR108
173_CR56
173_CR58
JL López (173_CR50) 1999; 6
FWJ Olver (173_CR77) 1991; 22
173_CR107
W Gautschi (173_CR24) 1967; 9
A Gil (173_CR28) 2006; 190
N Hale (173_CR39) 2013; 35
J Segura (173_CR88) 2008; 111
173_CR53
AB Olde Daalhuis (173_CR71) 2010; 8
X Wang (173_CR103) 2015; 258
A Glaser (173_CR36) 2007; 29
BA Kniehl (173_CR47) 2012; 854
C Ferreira (173_CR22) 2006; 197
N Michel (173_CR61) 2008; 178
KL Bell (173_CR8) 1980; 20
JL López (173_CR52) 2013; 39
FWJ Olver (173_CR74) 1993; 24
AB Olde Daalhuis (173_CR70) 2003; 1
173_CR40
NM Temme (173_CR92) 2015
173_CR35
G Chiocchia (173_CR13) 1989; 17
173_CR34
C Zhao (173_CR109) 2011; 166
173_CR37
YP Hsu (173_CR41) 1993; 4
FWJ Olver (173_CR76) 1965; 2
AK Ibrahim (173_CR43) 2008; 56
YL Luke (173_CR54) 1977; 11
V Pierro (173_CR81) 2002; 334
C Eckart (173_CR18) 1930; 35
A Gil (173_CR33) 2007; 76
T Schmelzer (173_CR86) 2007; 45
AB Olde Daalhuis (173_CR72) 1995; 2
D Torrieri (173_CR100) 2012; 60
173_CR5
173_CR6
JL Spouge (173_CR89) 1994; 31
173_CR3
CJ Lanczos (173_CR49) 1964; 1
173_CR9
NM Temme (173_CR96) 1983; 41
NM Temme (173_CR94) 2000; 121
RC Forrey (173_CR23) 1997; 137
DS Jones (173_CR44) 2001; 24
173_CR20
JL López (173_CR51) 2010; 233
173_CR10
J Abad (173_CR2) 1995; 5
NM Temme (173_CR95) 2007; 16
W Bühring (173_CR11) 1987; 18
FWJ Olver (173_CR78) 1980; 86
M Gavrila (173_CR27) 1967; 163
References_xml – volume: 5
  start-page: 466
  year: 1979
  end-page: 481
  ident: CR25
  article-title: A computational procedure for incomplete gamma functions
  publication-title: ACM Trans. Math. Softw.
  doi: 10.1145/355853.355863
– ident: CR97
– volume: 76
  start-page: 879
  year: 2007
  end-page: 901
  ident: CR14
  article-title: Transitory minimal solutions of hypergeometric recursions and pseudoconvergence of associated continued fractions
  publication-title: Math. Comput.
  doi: 10.1090/S0025-5718-07-01934-5
– volume: 281
  start-page: 2342
  year: 2008
  end-page: 2348
  ident: CR48
  article-title: Scintillation index of a stochastic electromagnetic beam propagating in random media
  publication-title: Opt. Commun.
  doi: 10.1016/j.optcom.2007.12.047
– ident: CR115
– volume: 258
  start-page: 282
  year: 2015
  end-page: 295
  ident: CR103
  article-title: Numerical methods for the mean exit time and escape probability of two-dimensional stochastic dynamical systems with non-Gaussian noises
  publication-title: Appl. Math. Comput.
– volume: 190
  start-page: 270
  year: 2006
  end-page: 286
  ident: CR28
  article-title: The ABC of hyper recursions
  publication-title: J. Comput. Appl. Math.
  doi: 10.1016/j.cam.2005.01.041
– ident: CR80
– volume: 163
  start-page: 147
  year: 1967
  end-page: 155
  ident: CR27
  article-title: Elastic scattering of photons by a hydrogen atom
  publication-title: Phys. Rev.
  doi: 10.1103/PhysRev.163.147
– volume: 148
  start-page: 717
  year: 2004
  end-page: 723
  ident: CR84
  article-title: Application of basic hypergeometric series
  publication-title: Appl. Math. Comput.
– volume: 29
  start-page: 1420
  year: 2007
  end-page: 1438
  ident: CR36
  article-title: A fast algorithm for the calculation of the roots of special functions
  publication-title: SIAM J. Sci. Comput.
  doi: 10.1137/06067016X
– volume: 6
  start-page: 21
  year: 1999
  end-page: 56
  ident: CR16
  article-title: Asymptotic approximations for the Jacobi and ultraspherical polynomials, and related functions
  publication-title: Methods Appl. Anal.
– volume: 36
  start-page: 247
  year: 2004
  end-page: 264
  ident: CR60
  article-title: Numerical representations of the incomplete gamma function of complex-valued argument
  publication-title: Numer. Algorithms
  doi: 10.1023/B:NUMA.0000040063.91709.58
– volume: 86
  start-page: 213
  year: 1980
  end-page: 234
  ident: CR78
  article-title: Whittaker functions with both parameters large: uniform approximations in terms of parabolic cylinder functions
  publication-title: Proc. Royal Soc. Edinb. Sect. A
  doi: 10.1017/S0308210500012130
– ident: CR57
– volume: 1
  start-page: 111
  year: 2003
  end-page: 120
  ident: CR69
  article-title: Uniform asymptotic expansions for hypergeometric functions with large parameters I
  publication-title: Anal. Appl.
  doi: 10.1142/S0219530503000028
– ident: CR112
– ident: CR85
– ident: CR5
– volume: 111
  start-page: 109
  year: 2008
  end-page: 119
  ident: CR88
  article-title: Numerically satisfactory solutions of Kummer recurrence relations
  publication-title: Numer. Math.
  doi: 10.1007/s00211-008-0175-5
– volume: 2
  start-page: 2098
  year: 1995
  end-page: 2109
  ident: CR59
  article-title: A dispersion function for plasmas containing superthermal particles
  publication-title: Phys. Plasmas
  doi: 10.1063/1.871296
– volume: 5
  start-page: 74
  year: 1995
  end-page: 76
  ident: CR2
  article-title: Computation of the regular confluent hypergeometric function
  publication-title: The Mathematica Journal
– ident: CR66
– ident: CR91
– volume: 60
  start-page: 3509
  year: 2012
  end-page: 3518
  ident: CR100
  article-title: The outage probability of a finite ad hoc network in Nakagami fading
  publication-title: IEEE Trans. Commun.
  doi: 10.1109/TCOMM.2012.081512.110530
– volume: 31
  start-page: 1194
  year: 2011
  end-page: 1216
  ident: CR32
  article-title: Fast and accurate computation of the Weber parabolic cylinder function W(a,x)
  publication-title: IMA J. Numer. Anal.
  doi: 10.1093/imanum/drq012
– volume: 184
  start-page: 1973
  year: 2013
  end-page: 1980
  ident: CR42
  article-title: NumExp: Numerical epsilon expansion of hypergeometric functions
  publication-title: Comput. Phys. Commun.
  doi: 10.1016/j.cpc.2013.03.016
– volume: 2
  start-page: 225
  year: 1965
  end-page: 243
  ident: CR76
  article-title: On the asymptotic solution of second-order differential equations having an irregular singularity of rank one, with an application to Whittaker functions
  publication-title: J. Soc. Indust. Appl. Math. Ser. B (Numerical Analysis)
  doi: 10.1137/0702017
– ident: CR10
– volume: 46
  start-page: 653
  year: 2006
  end-page: 670
  ident: CR101
  article-title: Talbot quadratures and rational approximations BIT
  publication-title: Numer. Anal.
– ident: CR6
– volume: 39
  start-page: 271
  year: 1987
  end-page: 279
  ident: CR4
  article-title: Numerical computation of Tricomi’s psi function by the trapezoidal rule
  publication-title: Computing
  doi: 10.1007/BF02309560
– volume: 166
  start-page: 1076
  year: 2011
  end-page: 1079
  ident: CR109
  article-title: An exact solution for electroosmosis of non-Newtonian fluids in microchannels
  publication-title: J. Non-Newtonian Fluid Mech.
  doi: 10.1016/j.jnnfm.2011.05.006
– volume: 196
  start-page: 799
  year: 2014
  end-page: 814
  ident: CR102
  article-title: Detecting and measuring selection from gene frequency data
  publication-title: Genetics
  doi: 10.1534/genetics.113.152991
– volume: 2
  start-page: 116
  year: 1918
  end-page: 148
  ident: CR104
  article-title: The harmonic functions associated with the parabolic cylinder
  publication-title: Proc. Lond. Math. Soc.
  doi: 10.1112/plms/s2-17.1.116
– ident: CR108
– volume: 21
  start-page: 241
  year: 1990
  end-page: 261
  ident: CR99
  article-title: Uniform asymptotics for a class of integrals in terms of modified Bessel functions, with application to confluent hypergeometric functions
  publication-title: SIAM J. Math. Anal.
  doi: 10.1137/0521013
– volume: 17
  start-page: 13
  year: 1989
  end-page: 23
  ident: CR13
  article-title: A new transformation for computing hypergeometric series and the exact evaluation of the transonic adiabatic flow over a smooth bump
  publication-title: Comput. Fluids
  doi: 10.1016/0045-7930(89)90004-2
– ident: CR3
– volume: 178
  start-page: 535
  year: 2008
  end-page: 551
  ident: CR61
  article-title: Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl–Teller–Ginocchio potential wave functions
  publication-title: Comput. Phys. Commun.
  doi: 10.1016/j.cpc.2007.11.007
– volume: 31
  start-page: 931
  year: 1994
  end-page: 944
  ident: CR89
  article-title: Computation of the gamma, digamma, and trigamma functions
  publication-title: SIAM J. Numer. Anal.
  doi: 10.1137/0731050
– ident: CR114
– volume: 18
  start-page: 884
  year: 1987
  end-page: 889
  ident: CR11
  article-title: An analytic continuation of the hypergeometric series
  publication-title: SIAM J. Math. Anal.
  doi: 10.1137/0518066
– ident: CR55
– volume: 35
  start-page: A652
  year: 2013
  end-page: A674
  ident: CR39
  article-title: Fast and accurate computation of Gauss–Legendre and Gauss–Jacobi quadrature nodes and weights
  publication-title: SIAM J. Sci. Comput.
  doi: 10.1137/120889873
– ident: CR83
– volume: 44
  start-page: 2342
  year: 2006
  end-page: 2362
  ident: CR105
  article-title: Optimizing Talbot’s contours for the inversion of the Laplace transform
  publication-title: SIAM J. Numer. Anal.
  doi: 10.1137/050625837
– volume: 137
  start-page: 79
  year: 1997
  end-page: 100
  ident: CR23
  article-title: Computing the hypergeometric function
  publication-title: J. Comput. Phys.
  doi: 10.1006/jcph.1997.5794
– volume: 9
  start-page: 24
  year: 1967
  end-page: 82
  ident: CR24
  article-title: Computational aspects of three-term recurrence relations
  publication-title: SIAM Rev.
  doi: 10.1137/1009002
– volume: 18
  start-page: 345
  year: 1992
  end-page: 349
  ident: CR64
  article-title: Algorithm 707: CONHYP: A numerical evaluator of the confluent hypergeometric function for complex arguments of large magnitudes
  publication-title: ACM Trans. Math. Softw.
  doi: 10.1145/131766.131774
– volume: 20
  start-page: 744
  year: 1989
  end-page: 760
  ident: CR17
  article-title: Uniform asymptotic expansions for Whittaker’s confluent hypergeometric function
  publication-title: SIAM J. Math. Anal.
  doi: 10.1137/0520052
– volume: 854
  start-page: 841
  year: 2012
  end-page: 852
  ident: CR47
  article-title: Finding new relationships between hypergeometric functions by evaluating Feynman integrals
  publication-title: Nuclear Phys. B
  doi: 10.1016/j.nuclphysb.2011.09.015
– volume: 30
  start-page: 159
  year: 2004
  end-page: 164
  ident: CR30
  article-title: Algorithm 831: Modified Bessel functions of imaginary order and positive argument
  publication-title: Trans. Math. Softw.
  doi: 10.1145/992200.992204
– volume: 45
  start-page: 558
  year: 2007
  end-page: 571
  ident: CR86
  article-title: Computing the gamma function using contour integrals and rational approximations
  publication-title: SIAM J. Numer. Anal.
  doi: 10.1137/050646342
– volume: 24
  start-page: 756
  year: 1993
  end-page: 767
  ident: CR74
  article-title: Exponentially-improved asymptotic solutions of ordinary differential equations I: the confluent hypergeometric function
  publication-title: SIAM J. Math. Anal.
  doi: 10.1137/0524046
– volume: 28
  start-page: 325
  year: 2002
  end-page: 336
  ident: CR29
  article-title: Algorithm 819: AIZ, BIZ: Two Fortran 77 routines for the computation of complex Airy functions
  publication-title: Trans. Math. Softw.
  doi: 10.1145/569147.569150
– volume: 63
  start-page: 066117
  year: 2001
  ident: CR79
  article-title: Epidemic dynamics and endemic states in complex networks.
  publication-title: Phys. Rev. E
  doi: 10.1103/PhysRevE.63.066117
– ident: CR87
– volume: 24
  start-page: 369
  year: 2001
  end-page: 389
  ident: CR44
  article-title: Asymptotics of the hypergeometric function
  publication-title: Math. Methods Appl. Sci.
  doi: 10.1002/mma.208
– volume: 41
  start-page: 43
  year: 1983
  end-page: 82
  ident: CR96
  article-title: The numerical computation of the confluent hypergeometric function U(a,b,z)
  publication-title: Numer. Math.
  doi: 10.1007/BF01396306
– ident: CR35
– ident: CR111
– volume: 22
  start-page: 1475
  year: 1991
  end-page: 1489
  ident: CR77
  article-title: Uniform, exponentially improved, asymptotic expansions for the confluent hypergeometric function and other integral transforms
  publication-title: SIAM J. Math. Anal.
  doi: 10.1137/0522095
– ident: CR58
– volume: 1
  start-page: 86
  year: 1964
  end-page: 96
  ident: CR49
  article-title: A precision approximation of the gamma function
  publication-title: SIAM J. Numer. Anal. Ser. B
– volume: 334
  start-page: 855
  year: 2002
  end-page: 858
  ident: CR81
  article-title: Computation of hypergeometric functions for gravitationally radiating binary stars
  publication-title: Mon. Not. Royal Astron. Soc.
  doi: 10.1046/j.1365-8711.2002.05557.x
– volume: 22
  start-page: 32353
  year: 2014
  end-page: 32360
  ident: CR19
  article-title: Intuitive model for the scintillations of a partially coherent beam
  publication-title: Opt. Express
  doi: 10.1364/OE.22.032353
– year: 2015
  ident: CR92
  publication-title: Asymptotic Methods for Integrals
– volume: 153
  start-page: 441
  year: 2003
  end-page: 462
  ident: CR93
  article-title: Large parameter cases of the Gauss hypergeometric function
  publication-title: J. Comput. Appl. Math.
  doi: 10.1016/S0377-0427(02)00627-1
– volume: 49
  start-page: 203
  year: 1992
  end-page: 216
  ident: CR68
  article-title: Hyperasymptotic expansions of confluent hypergeometric functions
  publication-title: IMA J. Appl. Math.
  doi: 10.1093/imamat/49.3.203
– volume: 233
  start-page: 1570
  year: 2010
  end-page: 1576
  ident: CR51
  article-title: The confluent hypergeometric functions M(a,b;z) and U(a,b;z) for large b and z
  publication-title: J. Comput. Appl. Math.
  doi: 10.1016/j.cam.2009.02.072
– volume: 76
  start-page: 1341
  year: 2007
  end-page: 1356
  ident: CR106
  article-title: Parabolic and hyperbolic contours for computing the Bromwich integral
  publication-title: Math. Comput.
  doi: 10.1090/S0025-5718-07-01945-X
– volume: 34
  start-page: A2965
  year: 2012
  end-page: A2981
  ident: CR31
  article-title: Efficient and accurate algorithms for the computation and inversion of the incomplete gamma function ratios
  publication-title: SIAM J. Sci. Comput.
  doi: 10.1137/120872553
– ident: CR116
– volume: 139
  start-page: 173
  year: 2002
  end-page: 187
  ident: CR26
  article-title: Gauss quadrature approximations to hypergeometric and confluent hypergeometric functions
  publication-title: J. Comput. Appl. Math.
  doi: 10.1016/S0377-0427(01)00383-1
– volume: 45
  start-page: 35
  year: 1992
  end-page: 36
  ident: CR45
  article-title: On the evaluation of the Gauss hypergeometric function
  publication-title: Compt. Rendus l’Acad. Bulgare Sci.
– ident: CR9
– volume: 1
  start-page: 121
  year: 2003
  end-page: 128
  ident: CR70
  article-title: Uniform asymptotic expansions for hypergeometric functions with large parameters II
  publication-title: Anal. Appl.
  doi: 10.1142/S021953050300003X
– volume: 39
  start-page: 349
  year: 2013
  end-page: 365
  ident: CR52
  article-title: New series expansions of the Gauss hypergeometric function
  publication-title: Adv. Comput. Math.
  doi: 10.1007/s10444-012-9283-y
– volume: 39
  start-page: 193
  year: 1992
  end-page: 200
  ident: CR65
  article-title: Numerical evaluation of the confluent hypergeometric function for complex arguments of large magnitudes
  publication-title: J. Comput. Appl. Math.
  doi: 10.1016/0377-0427(92)90129-L
– volume: 187
  start-page: 195
  year: 2015
  end-page: 203
  ident: CR21
  article-title: Coulomb wave functions in momentum space
  publication-title: Comput. Phys. Commun.
  doi: 10.1016/j.cpc.2014.10.002
– volume: 2
  start-page: 173
  year: 1995
  end-page: 197
  ident: CR72
  article-title: Hyperasymptotic solutions of second-order linear differential equations I
  publication-title: Methods Appl. Anal.
– volume: 22
  start-page: 215
  year: 1978
  end-page: 223
  ident: CR98
  article-title: Uniform asymptotic expansions of confluent hypergeometric functions
  publication-title: J. Inst. Math. Appl.
  doi: 10.1093/imamat/22.2.215
– ident: CR37
– ident: CR53
– volume: 11
  start-page: 123
  year: 1977
  end-page: 151
  ident: CR54
  article-title: Algorithms for rational approximations for a confluent hypergeometric function
  publication-title: Utilitas Math.
– volume: 35
  start-page: 1303
  year: 1930
  end-page: 1309
  ident: CR18
  article-title: The penetration of a potential barrier by electrons
  publication-title: Phys. Rev.
  doi: 10.1103/PhysRev.35.1303
– ident: CR82
– ident: CR113
– volume: 76
  start-page: 1449
  year: 2007
  end-page: 1468
  ident: CR33
  article-title: Numerically satisfactory solutions of hypergeometric recursions
  publication-title: Math. Comput.
  doi: 10.1090/S0025-5718-07-01918-7
– volume: 23
  start-page: 221
  year: 1969
  end-page: 230
  ident: CR38
  article-title: Calculation of Gauss quadrature rules
  publication-title: Math. Comput.
  doi: 10.1090/S0025-5718-69-99647-1
– volume: 19
  start-page: 1249
  year: 1988
  end-page: 1251
  ident: CR12
  article-title: An analytic continuation formula for the generalized hypergeometric function
  publication-title: SIAM J. Math. Anal.
  doi: 10.1137/0519088
– ident: CR56
– volume: 90
  start-page: 179
  year: 2001
  end-page: 196
  ident: CR63
  article-title: Computing the confluent hypergeometric function, M(a,b,x)
  publication-title: Numer. Math.
  doi: 10.1007/s002110100285
– ident: CR40
– volume: 6
  start-page: 249
  year: 1999
  end-page: 256
  ident: CR50
  article-title: Asymptotic expansions of the Whittaker functions for large order parameter
  publication-title: Methods Appl. Anal.
– volume: 71
  start-page: 47
  year: 1992
  end-page: 55
  ident: CR7
  article-title: Computation of the Kummer functions and Whittaker functions by using Neumann type series expansions
  publication-title: Comput. Phys. Commun.
  doi: 10.1016/0010-4655(92)90071-6
– volume: 20
  start-page: 447
  year: 1980
  end-page: 458
  ident: CR8
  article-title: Coulomb functions (negative energies)
  publication-title: Comput. Phys. Commun.
  doi: 10.1016/0010-4655(80)90022-3
– volume: 4
  start-page: 805
  year: 1993
  end-page: 840
  ident: CR41
  article-title: Development of a Gaussian hypergeometric function code in complex domains
  publication-title: Int. J. Modern Phys. C
  doi: 10.1142/S0129183193000665
– volume: 16
  start-page: 379
  year: 2007
  end-page: 478
  ident: CR95
  article-title: Numerical aspects of special functions
  publication-title: Acta Numer.
  doi: 10.1017/S0962492906330012
– volume: 12
  start-page: 667
  year: 2014
  end-page: 710
  ident: CR46
  article-title: Uniform asymptotic expansions for hypergeometric functions with large parameters IV
  publication-title: Anal. Appl.
  doi: 10.1142/S0219530514500389
– volume: 33
  start-page: 413
  year: 1984
  end-page: 419
  ident: CR67
  article-title: COULN, a program for evaluating negative energy coulomb functions
  publication-title: Comput. Phys. Commun.
  doi: 10.1016/0010-4655(84)90146-2
– ident: CR73
– ident: CR90
– volume: 121
  start-page: 221
  year: 2000
  end-page: 246
  ident: CR94
  article-title: Numerical and asymptotic aspects of parabolic cylinder functions
  publication-title: J. Comput. Appl. Math.
  doi: 10.1016/S0377-0427(00)00347-2
– volume: 197
  start-page: 568
  year: 2006
  end-page: 577
  ident: CR22
  article-title: The Gauss hypergeometric function F(a,b;c;z) for large c
  publication-title: J. Comput. Appl. Math.
  doi: 10.1016/j.cam.2005.11.027
– volume: 71
  start-page: 111
  year: 1967
  end-page: 129
  ident: CR75
  article-title: Numerical solution of second order linear difference equations
  publication-title: J. Res. Nat. Bur. Stand. Sect. B
  doi: 10.6028/jres.071B.018
– volume: 101
  start-page: 237
  year: 1999
  end-page: 241
  ident: CR1
  article-title: Buchholz polynomials: A family of polynomials relating solutions of confluent hypergeometric and Bessel equations
  publication-title: J. Comput. Appl. Math.
  doi: 10.1016/S0377-0427(99)00226-5
– ident: CR34
– ident: CR110
– volume: 35
  start-page: 88
  year: 2009
  end-page: 103
  ident: CR15
  article-title: On modified asymptotic series involving confluent hypergeometric functions
  publication-title: Electron. Trans. Numer. Anal.
– volume: 8
  start-page: 199
  year: 2010
  end-page: 210
  ident: CR71
  article-title: Uniform asymptotic expansions for hypergeometric functions with large parameters III
  publication-title: Anal. Appl.
  doi: 10.1142/S0219530510001588
– ident: CR107
– ident: CR62
– ident: CR20
– volume: 56
  start-page: 1918
  year: 2008
  end-page: 1926
  ident: CR43
  article-title: Contiguous relations and their computations for 2F1 hypergeometric series
  publication-title: Comput. Math. Appl.
  doi: 10.1016/j.camwa.2008.04.018
– ident: 173_CR110
– volume: 76
  start-page: 879
  year: 2007
  ident: 173_CR14
  publication-title: Math. Comput.
  doi: 10.1090/S0025-5718-07-01934-5
– volume: 187
  start-page: 195
  year: 2015
  ident: 173_CR21
  publication-title: Comput. Phys. Commun.
  doi: 10.1016/j.cpc.2014.10.002
– ident: 173_CR34
  doi: 10.1137/1.9780898717822
– volume: 1
  start-page: 121
  year: 2003
  ident: 173_CR70
  publication-title: Anal. Appl.
  doi: 10.1142/S021953050300003X
– ident: 173_CR5
  doi: 10.1063/1.2939395
– volume: 163
  start-page: 147
  year: 1967
  ident: 173_CR27
  publication-title: Phys. Rev.
  doi: 10.1103/PhysRev.163.147
– volume: 35
  start-page: 88
  year: 2009
  ident: 173_CR15
  publication-title: Electron. Trans. Numer. Anal.
– volume: 36
  start-page: 247
  year: 2004
  ident: 173_CR60
  publication-title: Numer. Algorithms
  doi: 10.1023/B:NUMA.0000040063.91709.58
– volume: 90
  start-page: 179
  year: 2001
  ident: 173_CR63
  publication-title: Numer. Math.
  doi: 10.1007/s002110100285
– volume: 166
  start-page: 1076
  year: 2011
  ident: 173_CR109
  publication-title: J. Non-Newtonian Fluid Mech.
  doi: 10.1016/j.jnnfm.2011.05.006
– ident: 173_CR53
  doi: 10.1090/psapm/048/1314844
– volume: 16
  start-page: 379
  year: 2007
  ident: 173_CR95
  publication-title: Acta Numer.
  doi: 10.1017/S0962492906330012
– volume: 24
  start-page: 369
  year: 2001
  ident: 173_CR44
  publication-title: Math. Methods Appl. Sci.
  doi: 10.1002/mma.208
– volume: 281
  start-page: 2342
  year: 2008
  ident: 173_CR48
  publication-title: Opt. Commun.
  doi: 10.1016/j.optcom.2007.12.047
– volume: 39
  start-page: 349
  year: 2013
  ident: 173_CR52
  publication-title: Adv. Comput. Math.
  doi: 10.1007/s10444-012-9283-y
– ident: 173_CR40
– volume: 17
  start-page: 13
  year: 1989
  ident: 173_CR13
  publication-title: Comput. Fluids
  doi: 10.1016/0045-7930(89)90004-2
– volume: 121
  start-page: 221
  year: 2000
  ident: 173_CR94
  publication-title: J. Comput. Appl. Math.
  doi: 10.1016/S0377-0427(00)00347-2
– ident: 173_CR20
– volume: 45
  start-page: 558
  year: 2007
  ident: 173_CR86
  publication-title: SIAM J. Numer. Anal.
  doi: 10.1137/050646342
– volume: 334
  start-page: 855
  year: 2002
  ident: 173_CR81
  publication-title: Mon. Not. Royal Astron. Soc.
  doi: 10.1046/j.1365-8711.2002.05557.x
– ident: 173_CR66
– ident: 173_CR113
– volume: 21
  start-page: 241
  year: 1990
  ident: 173_CR99
  publication-title: SIAM J. Math. Anal.
  doi: 10.1137/0521013
– volume: 31
  start-page: 1194
  year: 2011
  ident: 173_CR32
  publication-title: IMA J. Numer. Anal.
  doi: 10.1093/imanum/drq012
– volume: 45
  start-page: 35
  year: 1992
  ident: 173_CR45
  publication-title: Compt. Rendus l’Acad. Bulgare Sci.
– ident: 173_CR10
  doi: 10.1145/1145768.1145782
– volume: 18
  start-page: 345
  year: 1992
  ident: 173_CR64
  publication-title: ACM Trans. Math. Softw.
  doi: 10.1145/131766.131774
– volume: 12
  start-page: 667
  year: 2014
  ident: 173_CR46
  publication-title: Anal. Appl.
  doi: 10.1142/S0219530514500389
– ident: 173_CR83
– volume: 24
  start-page: 756
  year: 1993
  ident: 173_CR74
  publication-title: SIAM J. Math. Anal.
  doi: 10.1137/0524046
– volume: 20
  start-page: 447
  year: 1980
  ident: 173_CR8
  publication-title: Comput. Phys. Commun.
  doi: 10.1016/0010-4655(80)90022-3
– volume: 23
  start-page: 221
  year: 1969
  ident: 173_CR38
  publication-title: Math. Comput.
  doi: 10.1090/S0025-5718-69-99647-1
– volume: 46
  start-page: 653
  year: 2006
  ident: 173_CR101
  publication-title: Numer. Anal.
– volume: 44
  start-page: 2342
  year: 2006
  ident: 173_CR105
  publication-title: SIAM J. Numer. Anal.
  doi: 10.1137/050625837
– volume: 111
  start-page: 109
  year: 2008
  ident: 173_CR88
  publication-title: Numer. Math.
  doi: 10.1007/s00211-008-0175-5
– volume: 153
  start-page: 441
  year: 2003
  ident: 173_CR93
  publication-title: J. Comput. Appl. Math.
  doi: 10.1016/S0377-0427(02)00627-1
– volume: 41
  start-page: 43
  year: 1983
  ident: 173_CR96
  publication-title: Numer. Math.
  doi: 10.1007/BF01396306
– volume: 28
  start-page: 325
  year: 2002
  ident: 173_CR29
  publication-title: Trans. Math. Softw.
  doi: 10.1145/569147.569150
– ident: 173_CR87
  doi: 10.1007/978-1-4757-5443-8
– volume: 5
  start-page: 466
  year: 1979
  ident: 173_CR25
  publication-title: ACM Trans. Math. Softw.
  doi: 10.1145/355853.355863
– ident: 173_CR80
– volume: 60
  start-page: 3509
  year: 2012
  ident: 173_CR100
  publication-title: IEEE Trans. Commun.
  doi: 10.1109/TCOMM.2012.081512.110530
– volume: 86
  start-page: 213
  year: 1980
  ident: 173_CR78
  publication-title: Proc. Royal Soc. Edinb. Sect. A
  doi: 10.1017/S0308210500012130
– volume: 197
  start-page: 568
  year: 2006
  ident: 173_CR22
  publication-title: J. Comput. Appl. Math.
  doi: 10.1016/j.cam.2005.11.027
– volume: 137
  start-page: 79
  year: 1997
  ident: 173_CR23
  publication-title: J. Comput. Phys.
  doi: 10.1006/jcph.1997.5794
– volume: 5
  start-page: 74
  year: 1995
  ident: 173_CR2
  publication-title: The Mathematica Journal
– ident: 173_CR112
  doi: 10.31855/07868f86-19f
– ident: 173_CR115
– volume: 34
  start-page: A2965
  year: 2012
  ident: 173_CR31
  publication-title: SIAM J. Sci. Comput.
  doi: 10.1137/120872553
– ident: 173_CR91
– volume: 22
  start-page: 215
  year: 1978
  ident: 173_CR98
  publication-title: J. Inst. Math. Appl.
  doi: 10.1093/imamat/22.2.215
– ident: 173_CR37
– ident: 173_CR107
– volume: 56
  start-page: 1918
  year: 2008
  ident: 173_CR43
  publication-title: Comput. Math. Appl.
  doi: 10.1016/j.camwa.2008.04.018
– ident: 173_CR85
  doi: 10.1145/236869.237088
– ident: 173_CR6
  doi: 10.1017/CBO9781107325937
– volume: 258
  start-page: 282
  year: 2015
  ident: 173_CR103
  publication-title: Appl. Math. Comput.
  doi: 10.1016/j.amc.2015.01.117
– volume: 2
  start-page: 173
  year: 1995
  ident: 173_CR72
  publication-title: Methods Appl. Anal.
  doi: 10.4310/MAA.1995.v2.n2.a4
– volume: 1
  start-page: 111
  year: 2003
  ident: 173_CR69
  publication-title: Anal. Appl.
  doi: 10.1142/S0219530503000028
– volume: 190
  start-page: 270
  year: 2006
  ident: 173_CR28
  publication-title: J. Comput. Appl. Math.
  doi: 10.1016/j.cam.2005.01.041
– ident: 173_CR57
– volume: 6
  start-page: 249
  year: 1999
  ident: 173_CR50
  publication-title: Methods Appl. Anal.
  doi: 10.4310/MAA.1999.v6.n2.a8
– volume: 178
  start-page: 535
  year: 2008
  ident: 173_CR61
  publication-title: Comput. Phys. Commun.
  doi: 10.1016/j.cpc.2007.11.007
– volume: 9
  start-page: 24
  year: 1967
  ident: 173_CR24
  publication-title: SIAM Rev.
  doi: 10.1137/1009002
– volume: 184
  start-page: 1973
  year: 2013
  ident: 173_CR42
  publication-title: Comput. Phys. Commun.
  doi: 10.1016/j.cpc.2013.03.016
– volume: 19
  start-page: 1249
  year: 1988
  ident: 173_CR12
  publication-title: SIAM J. Math. Anal.
  doi: 10.1137/0519088
– ident: 173_CR108
– ident: 173_CR114
– ident: 173_CR73
– volume: 33
  start-page: 413
  year: 1984
  ident: 173_CR67
  publication-title: Comput. Phys. Commun.
  doi: 10.1016/0010-4655(84)90146-2
– volume: 20
  start-page: 744
  year: 1989
  ident: 173_CR17
  publication-title: SIAM J. Math. Anal.
  doi: 10.1137/0520052
– ident: 173_CR58
– ident: 173_CR82
– volume: 30
  start-page: 159
  year: 2004
  ident: 173_CR30
  publication-title: Trans. Math. Softw.
  doi: 10.1145/992200.992204
– volume: 39
  start-page: 271
  year: 1987
  ident: 173_CR4
  publication-title: Computing
  doi: 10.1007/BF02309560
– volume: 8
  start-page: 199
  year: 2010
  ident: 173_CR71
  publication-title: Anal. Appl.
  doi: 10.1142/S0219530510001588
– ident: 173_CR35
  doi: 10.1145/2049662.2049668
– volume: 71
  start-page: 47
  year: 1992
  ident: 173_CR7
  publication-title: Comput. Phys. Commun.
  doi: 10.1016/0010-4655(92)90071-6
– volume: 139
  start-page: 173
  year: 2002
  ident: 173_CR26
  publication-title: J. Comput. Appl. Math.
  doi: 10.1016/S0377-0427(01)00383-1
– volume: 854
  start-page: 841
  year: 2012
  ident: 173_CR47
  publication-title: Nuclear Phys. B
  doi: 10.1016/j.nuclphysb.2011.09.015
– ident: 173_CR55
– volume: 49
  start-page: 203
  year: 1992
  ident: 173_CR68
  publication-title: IMA J. Appl. Math.
  doi: 10.1093/imamat/49.3.203
– volume: 6
  start-page: 21
  year: 1999
  ident: 173_CR16
  publication-title: Methods Appl. Anal.
  doi: 10.4310/MAA.1999.v6.n3.a2
– ident: 173_CR62
– volume: 2
  start-page: 2098
  year: 1995
  ident: 173_CR59
  publication-title: Phys. Plasmas
  doi: 10.1063/1.871296
– volume: 22
  start-page: 1475
  year: 1991
  ident: 173_CR77
  publication-title: SIAM J. Math. Anal.
  doi: 10.1137/0522095
– ident: 173_CR56
  doi: 10.1016/B978-0-12-459950-5.50006-7
– volume: 29
  start-page: 1420
  year: 2007
  ident: 173_CR36
  publication-title: SIAM J. Sci. Comput.
  doi: 10.1137/06067016X
– volume: 2
  start-page: 225
  year: 1965
  ident: 173_CR76
  publication-title: J. Soc. Indust. Appl. Math. Ser. B (Numerical Analysis)
  doi: 10.1137/0702017
– ident: 173_CR116
– ident: 173_CR90
– ident: 173_CR97
  doi: 10.1002/9781118032572
– volume: 22
  start-page: 32353
  year: 2014
  ident: 173_CR19
  publication-title: Opt. Express
  doi: 10.1364/OE.22.032353
– volume: 196
  start-page: 799
  year: 2014
  ident: 173_CR102
  publication-title: Genetics
  doi: 10.1534/genetics.113.152991
– volume: 76
  start-page: 1341
  year: 2007
  ident: 173_CR106
  publication-title: Math. Comput.
  doi: 10.1090/S0025-5718-07-01945-X
– volume: 31
  start-page: 931
  year: 1994
  ident: 173_CR89
  publication-title: SIAM J. Numer. Anal.
  doi: 10.1137/0731050
– volume: 4
  start-page: 805
  year: 1993
  ident: 173_CR41
  publication-title: Int. J. Modern Phys. C
  doi: 10.1142/S0129183193000665
– ident: 173_CR9
– volume: 101
  start-page: 237
  year: 1999
  ident: 173_CR1
  publication-title: J. Comput. Appl. Math.
  doi: 10.1016/S0377-0427(99)00226-5
– ident: 173_CR3
– volume: 11
  start-page: 123
  year: 1977
  ident: 173_CR54
  publication-title: Utilitas Math.
– volume: 35
  start-page: A652
  year: 2013
  ident: 173_CR39
  publication-title: SIAM J. Sci. Comput.
  doi: 10.1137/120889873
– ident: 173_CR111
– volume: 1
  start-page: 86
  year: 1964
  ident: 173_CR49
  publication-title: SIAM J. Numer. Anal. Ser. B
– volume: 76
  start-page: 1449
  year: 2007
  ident: 173_CR33
  publication-title: Math. Comput.
  doi: 10.1090/S0025-5718-07-01918-7
– volume: 148
  start-page: 717
  year: 2004
  ident: 173_CR84
  publication-title: Appl. Math. Comput.
  doi: 10.1016/S0096-3003(02)00930-X
– volume: 71
  start-page: 111
  year: 1967
  ident: 173_CR75
  publication-title: J. Res. Nat. Bur. Stand. Sect. B
  doi: 10.6028/jres.071B.018
– volume: 18
  start-page: 884
  year: 1987
  ident: 173_CR11
  publication-title: SIAM J. Math. Anal.
  doi: 10.1137/0518066
– volume: 63
  start-page: 066117
  year: 2001
  ident: 173_CR79
  publication-title: Phys. Rev. E
  doi: 10.1103/PhysRevE.63.066117
– volume: 39
  start-page: 193
  year: 1992
  ident: 173_CR65
  publication-title: J. Comput. Appl. Math.
  doi: 10.1016/0377-0427(92)90129-L
– volume: 2
  start-page: 116
  year: 1918
  ident: 173_CR104
  publication-title: Proc. Lond. Math. Soc.
  doi: 10.1112/plms/s2-17.1.116
– volume: 35
  start-page: 1303
  year: 1930
  ident: 173_CR18
  publication-title: Phys. Rev.
  doi: 10.1103/PhysRev.35.1303
– volume-title: Asymptotic Methods for Integrals
  year: 2015
  ident: 173_CR92
– volume: 233
  start-page: 1570
  year: 2010
  ident: 173_CR51
  publication-title: J. Comput. Appl. Math.
  doi: 10.1016/j.cam.2009.02.072
SSID ssj0010027
Score 2.4039133
Snippet The two most commonly used hypergeometric functions are the confluent hypergeometric function and the Gauss hypergeometric function. We review the available...
SourceID proquest
crossref
springer
SourceType Aggregation Database
Enrichment Source
Index Database
Publisher
StartPage 821
SubjectTerms Algebra
Algorithms
Applied mathematics
Asymptotic methods
Asymptotic series
Computation
Computer Science
Differential equations
Hypergeometric functions
Numeric Computing
Numerical Analysis
Numerical methods
Original Paper
Parameters
Quadratures
Recommender systems
Theory of Computation
SummonAdditionalLinks – databaseName: Engineering Database
  dbid: M7S
  link: http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV09T8MwED1BYYCBQgFRKMgDE8giiZPGmRBCFBYqJEAqU5TYDiCVpjQpv5-z4zQCiS6s8Yes3Nn3zne-B3AqcXUuWgLqek5K_VR6NEJgTwPhJixx_UAJacgmwuGQj0bRg71wK2xaZX0mmoNa5kLfkV94kcsRbAcuu5x-Us0apaOrlkJjFdZ0lQTXpO49LqII2ucy0U48iRHp8DqqaZ7Ood-j09a0Px0y6vy0Sw3Y_BUfNWZn0P7vgrdhywJOclVpyA6sqEkH2hZ8Eru1iw5s3i8KuBa78DKcV6GcMak4pguC6JZgDyIMD4QRKMkz-6miOilJMpHkNpkXBXlDB3f2qvIPTdkliLafRsX34Hlw83R9Ry0LAxW4n0vKmGTKizLhB4l0Mikz7mdJ6CWIBFADvEylKbb1nbQfqlCmUYoejUIolfIsxP_L9qE1ySfqAEjIcQTPdAX9xBci1Niojy4XKoVyOWddcGoZxMKWKNdMGeO4Ka6sxRbrtDQtttjpwtliyLSqz7Gsc68WVWy3ahE3curCeS3spvnPyQ6XT3YEG57WO5Ou1oNWOZurY1gXX-V7MTsxevoN5QHt5w
  priority: 102
  providerName: ProQuest
Title Numerical methods for the computation of the confluent and Gauss hypergeometric functions
URI https://link.springer.com/article/10.1007/s11075-016-0173-0
https://www.proquest.com/docview/2918606513
Volume 74
WOSCitedRecordID wos000395033800009&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D
hasFullText 1
inHoldings 1
isFullTextHit
isPrint
journalDatabaseRights – providerCode: PRVAVX
  databaseName: SpringerLINK Contemporary 1997-Present
  customDbUrl:
  eissn: 1572-9265
  dateEnd: 99991231
  omitProxy: false
  ssIdentifier: ssj0010027
  issn: 1017-1398
  databaseCode: RSV
  dateStart: 19970101
  isFulltext: true
  titleUrlDefault: https://link.springer.com/search?facet-content-type=%22Journal%22
  providerName: Springer Nature
link http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV3dS8MwED90-qAPTqfidJY8-KQE-rUlfVTZFMQyNpX5VNokVWF2sm7-_V7SdkNRQV_z1ZK75H7HXe4HcCLx7xy0BNRx7YT6iXRpgMCetoUTe7Hjt5WQhmyChSEfjYJ--Y47r7Ldq5CkuamXj93QU9GJZtoDZh5FP30NrR3XfA2D4cMidKAdLRPixOsX4Q2vQpnfLfHZGC0R5pegqLE1vfq__nIbtkpoSc4LXdiBFZU1oF7CTFIe4hybKiaHqq0Bm7eL6q35LjyG8yKOMyYFwXROENoSHEGEmWqkSSZp2VTwnMxInElyFc_znDyjdzt9UpNXzdcliDaeRr_34L7Xvbu8piUFAxV4mGfU86Sn3CAVfjuWdiplyv00Zm6MMADF76YqSbCvYycdpphMggTdGYU4KuEpw03w9qGWTTJ1AIRxnMFTXT4_9oVgGhh10N9CjVAO514T7EoWkSjrk2uajHG0rKys9zbSOWl6byO7CaeLKW9FcY7fBrcqAUflOc0jN3A4unBtBz9_Vgl02f3jYod_Gn0EG65WR5O61oLabDpXx7Au3mcv-dSCtYtu2B9YsHrDqKWTT4eWUekP0bLs1w
linkProvider Springer Nature
linkToHtml http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMw1V3Pb9MwFH7qChJwoDCG6CjgA1yYrCV2WjsHNE1AadWt4tBJ45QltrMhjXY0LdP-qf2NvOckrZjEbjvsGv9QHH9-_l7esz-A9xbfLsSdgIciyHiUWcFjJPa8a8JUpmHUdcZ6sQk1Huvj4_h7A67rszCUVlnbRG-o7czQP_JdEYcayXY3lHsXvzmpRlF0tZbQKGExcleX6LIVn4ZfcH4_CNH_Ovk84JWqADeIzwWX0kon4txE3dQGubW5jvJUiRR3NhyRyF2WYVkvyHrKKZvFGTJ0h9Qg07lCgiSx3w14EEmtaF2NFF9FLcjH89FVtPzIrHQdRfVH9dDPojQ58t-V5MG_--Ca3N6Ix_ptrt-6bx_oGTytCDXbL1fAc2i46Sa0KnLNKtNVbMKTw9UFtcUL-DFelqGqc1ZqaBcM2TvDGsx4nQsPWDbLq0ellMuCpVPLvqXLomBn6MDPT93sF0mSGUb8wC_hLTi6k-G-hOZ0NnWvgCmNLXROCgFpZIwi7tdDlxJB70KtZRuCes4TU13BTkog58n68miCSUJpdwSTJGjDx1WTi_L-kdsqd2poJJUpKpI1LtqwU4NrXfzfzrZv7-wdPBpMDg-Sg-F49BoeC8K8T83rQHMxX7o38ND8Wfws5m_9GmFwcteY-wuJxU0J
linkToPdf http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwpV1LS8NAEF5ERfRgtSpWq-7BkxKaZ7M5iloVNRR8UE9hsw8Valqa1N_v7GbToKggXveVZWc2833M7AxChxx254AlsBzXTi0_5a4VAbC3AuZQjzp-IBjXxSbCOCaDQdQ3dU7zKtq9ckmWbxpUlqas6Iy57NQP34C1qKAzxYZDzwLOvuCrOHpF1-8eZ24ERbq0uxN-xQB1SOXW_G6Jz4apRptfHKTa7vQa_97xGlo1kBOflDqyjuZE1kQNAz-xudw5NFUVHqq2Jlq5nWV1zTfQUzwt_TtDXBaezjFAXgwjMNNTtZTxSJqmsv5JgWnG8QWd5jl-AdY7eRajN1XHi2FlVLXeb6KH3vn96aVlSjNYDC55YXke94QbSeYHlNuSc0l8SUOXAjwAtXClSFPo69ppNxQhT6MUaI4AfJUSGcIheFtoPhtlYhvhkMAMIlVafeozFirA1AUeBpoiHEK8FrIruSTM5C1X5TOGSZ1xWZ1tomLV1NkmdgsdzaaMy6Qdvw1uV8JOzP3NEzdyCFC7wIHPH1fCrbt_XGznT6MP0FL_rJfcXMXXu2jZVVqqo9vaaL6YTMUeWmTvxWs-2dda_QET4vU9
openUrl ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=Numerical+methods+for+the+computation+of+the+confluent+and+Gauss+hypergeometric+functions&rft.jtitle=Numerical+algorithms&rft.au=Pearson%2C+John+W.&rft.au=Olver%2C+Sheehan&rft.au=Porter%2C+Mason+A.&rft.date=2017-03-01&rft.issn=1017-1398&rft.eissn=1572-9265&rft.volume=74&rft.issue=3&rft.spage=821&rft.epage=866&rft_id=info:doi/10.1007%2Fs11075-016-0173-0&rft.externalDBID=n%2Fa&rft.externalDocID=10_1007_s11075_016_0173_0
thumbnail_l http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=1017-1398&client=summon
thumbnail_m http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=1017-1398&client=summon
thumbnail_s http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=1017-1398&client=summon