Properties of the Appell–Lerch function (I)

A number of equations involving the Appell–Lerch function, μ , are derived. Emphasis is placed on equations which are analogous to certain linear relations which exist between theta functions, as well as equations which make explicit the symmetry μ ( x , y ) = μ ( y , x ) . Two classical formulae wh...

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Vydáno v:The Ramanujan journal Ročník 57; číslo 1; s. 291 - 367
Hlavní autor: Bradley-Thrush, J. G.
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.01.2022
Springer Nature B.V
Témata:
Combinatorics
Field Theory and Polynomials
Fourier Analysis
Functions of a Complex Variable
Mathematical analysis
Mathematics
Mathematics and Statistics
Number Theory
Series expansion
11F27
Basic hypergeometric functions
33D65
Theta functions
11B65
30D05
Appell–Lerch series
33D15
ISSN:1382-4090, 1572-9303
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Shrnutí:A number of equations involving the Appell–Lerch function, μ , are derived. Emphasis is placed on equations which are analogous to certain linear relations which exist between theta functions, as well as equations which make explicit the symmetry μ ( x , y ) = μ ( y , x ) . Two classical formulae which appear in the work of Halphen and Lerch are examined in relation to the more recent work of McIntosh and that of Zagier. Four proofs of Halphen’s formula are presented. A related bibasic series is then considered and a result is obtained which generalizes some double series expansions, originally due to Garvan, for the universal mock theta functions.
Bibliografie:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:1382-4090
1572-9303
DOI:10.1007/s11139-021-00445-4