Monogenic pseudo-complex power functions and their applications
The use of a non‐commutative algebra in hypercomplex function theory requires a large variety of different representations of polynomials suitably adapted to the solution of different concrete problems. Naturally arises the question of their relationships and the advantages or disadvantages of diffe...
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| Published in: | Mathematical methods in the applied sciences Vol. 37; no. 12; pp. 1723 - 1735 |
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01.08.2014
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| ISSN: | 0170-4214, 1099-1476 |
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| Abstract | The use of a non‐commutative algebra in hypercomplex function theory requires a large variety of different representations of polynomials suitably adapted to the solution of different concrete problems. Naturally arises the question of their relationships and the advantages or disadvantages of different types of polynomials. In this sense, the present paper investigates the intrinsic relationship between two different types of monogenic Appell polynomials. Several authors payed attention to the construction of complete sets of monogenic Appell polynomials, orthogonal with respect to a certain inner product, and used them advantageously for the study of problems in 3D‐elasticity and other problems. Our goal is to show that, as consequence of the binomial nature of those generalized Appell polynomials, their inner structure is determined by interesting combinatorial relations in which the central binomial coefficients play a special role. As a byproduct of own interest, a Riordan–Sofo type binomial identity is also proved. Copyright © 2013 John Wiley & Sons, Ltd. |
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| AbstractList | The use of a non-commutative algebra in hypercomplex function theory requires a large variety of different representations of polynomials suitably adapted to the solution of different concrete problems. Naturally arises the question of their relationships and the advantages or disadvantages of different types of polynomials. In this sense, the present paper investigates the intrinsic relationship between two different types of monogenic Appell polynomials. Several authors payed attention to the construction of complete sets of monogenic Appell polynomials, orthogonal with respect to a certain inner product, and used them advantageously for the study of problems in 3D-elasticity and other problems. Our goal is to show that, as consequence of the binomial nature of those generalized Appell polynomials, their inner structure is determined by interesting combinatorial relations in which the central binomial coefficients play a special role. As a byproduct of own interest, a Riordan-Sofo type binomial identity is also proved. Copyright copyright 2013 John Wiley & Sons, Ltd. The use of a non-commutative algebra in hypercomplex function theory requires a large variety of different representations of polynomials suitably adapted to the solution of different concrete problems. Naturally arises the question of their relationships and the advantages or disadvantages of different types of polynomials. In this sense, the present paper investigates the intrinsic relationship between two different types of monogenic Appell polynomials. Several authors payed attention to the construction of complete sets of monogenic Appell polynomials, orthogonal with respect to a certain inner product, and used them advantageously for the study of problems in 3D-elasticity and other problems. Our goal is to show that, as consequence of the binomial nature of those generalized Appell polynomials, their inner structure is determined by interesting combinatorial relations in which the central binomial coefficients play a special role. As a byproduct of own interest, a Riordan-Sofo type binomial identity is also proved. Copyright © 2013 John Wiley & Sons, Ltd. [PUBLICATION ABSTRACT] The use of a non‐commutative algebra in hypercomplex function theory requires a large variety of different representations of polynomials suitably adapted to the solution of different concrete problems. Naturally arises the question of their relationships and the advantages or disadvantages of different types of polynomials. In this sense, the present paper investigates the intrinsic relationship between two different types of monogenic Appell polynomials. Several authors payed attention to the construction of complete sets of monogenic Appell polynomials, orthogonal with respect to a certain inner product, and used them advantageously for the study of problems in 3D‐elasticity and other problems. Our goal is to show that, as consequence of the binomial nature of those generalized Appell polynomials, their inner structure is determined by interesting combinatorial relations in which the central binomial coefficients play a special role. As a byproduct of own interest, a Riordan–Sofo type binomial identity is also proved. Copyright © 2013 John Wiley & Sons, Ltd. |
| Author | Irene Falcão, Maria Cruz, Carla Malonek, Helmuth R. |
| Author_xml | – sequence: 1 givenname: Carla surname: Cruz fullname: Cruz, Carla organization: Center for Research and Development in Mathematics and Applications, University of Aveiro, Aveiro, Portugal – sequence: 2 givenname: Maria surname: Irene Falcão fullname: Irene Falcão, Maria organization: Center for Research and Development in Mathematics and Applications, University of Aveiro, Aveiro, Portugal – sequence: 3 givenname: Helmuth R. surname: Malonek fullname: Malonek, Helmuth R. email: Correspondence to: Helmuth R.Malonek, Department of Mathematics, University of Aveiro, Aveiro, Portugal., hrmalon@ua.pt organization: Center for Research and Development in Mathematics and Applications, University of Aveiro, Aveiro, Portugal |
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| References | Delanghe R. On regular-analytic functions with values in a Clifford algebra. Mathematische Annalen 1970; 185: 91-111. Delanghe R, Lávička R, Souček V. The Gelfand-Tsetlin bases for Hodge-de Rham systems in Euclidean spaces. Mathematical Methods In The Applied Sciences 2012; 35(7):745-757. Moisil GrC, Théodoresco N. Fonctions holomorphes dans l'espace. Mathematica 1931; 5: 142-159. Lávička R. Canonical bases for sl(2, C)-modules of spherical monogenics in dimension 3. Archivum Mathematicum 2010; 46: 339-349. Malonek HR. Power series representation for monogenic functions in Rn+1 based on a permutational product. Complex Variables, Theory Applications 1990; 15: 181-191. Prodinger H. Knuth's old sum - a survey. European Association for Theoretical Computer Science Bulletin 1994; 54: 232-245. Fueter R. Analytische Funktionen einer Quaternionenvariablen. Commentarii Mathematici Helvetici 1932; 4: 9-20. Brackx F. Two powerful theorems in Clifford analysis. AIP Conference Proceedings 2010; 1281: 3-7. Lehmer HD. Interesting series involving the central binomial coefficient. American Mathematical Monthly 1985; 92: 449-457. Cação I, Falcão MI, Malonek HR. On generalized hypercomplex Laguerre-type exponentials and applications. Lecture Notes in Computer Science 2011; 6784: 271-286. Gürlebeck K, Malonek HR. A hypercomplex derivative of monogenic functions in Rn+1 and its applications. Complex Variables Theory Applications 1999; 39: 199-228. Malonek HR, de Almeida R. A note on a generalized Joukowski transformation. Applied Mathematics Letters 2010; 23(10):1174-1178. Appell P. Sur une classe de pôlynomes. Annals scientifiques de l'École Normal Supérieure 21 série 1880; 9(2):119-144. Cruz C, Falcão MI, Malonek HR. On pseudo-complex bases for monogenic polynomials. AIP Conference Proceedings 2012; 1493: 350-355. Faà di Bruno CF. Note sur une nouvelle formule, de calcul différentiel. Quarterly Journal of Pure and Applied Math 1857; 1: 359-360. Bock S, Gürlebeck K. On a generalized Appell system and monogenic power series. Mathematical Methods In The Applied Sciences 2010; 33(4):394-411. Sofo A. Computational Techniques for the Summation of Series, Kluwer: New York, 2003. Riordan J. Combinatorial Identities, John Wiley & Sons Inc.: New York, 1968. Johnson WP. The curious history of Faà di Bruno's formula. The Mathematical Association of America Monthly 2002; 109(3):217-234. Cruz C, Falcão MI, Malonek HR. 3D mappings by generalized joukowski transformation. Lecture Notes in Computer Science 2011; 6784: 358-373. Falcão MI, Malonek HR. On paravector valued homogeneous monogenic polynomials with binomial expansion. Advances in Applied Clifford Algebras 2012; 22(3):789-801. Fueter R. Funktionentheorie im Hyperkomplexen, Lecture notes written and supplemented by E. Bareiss, Math. Inst. Univ. Zürich. Universität Zürich: Herbstsemester,1948/49, 1949. Cação I, Falcão MI, Malonek HR. Laguerre derivative and monogenic Laguerre polynomials: an operational approach. Mathematical and Computer Modelling 2011; 53(5-6):1084-1094. Falcão MI, Malonek HR. Generalized exponentials through Appell sets in Rn+1 and Bessel functions. AIP Conference Procedings 2007; 936: 738-741. Sofo A. Closed form representation of binomial sums and series. Le Matematiche 1999; LIV: 175-186. Brackx F, Delanghe R, Sommen F. Clifford Analysis. Pitman 76: London, 1982. Gürlebeck K, Habetha K, Sprössig W. Holomorphic Functions in the Plane and n-Dimensional Space, Birkäuser Verlag: Basel, 2008. Cação I, Falcão MI, Malonek HR. Matrix representations of a special polynomial sequence in arbitrary dimension. Computational Methods and Function Theory 2012; 12(2):371-391. Krantz SG. Function Theory of Several Complex Variables, 2nd ed. American Mathematical Society, Providenc: RI, 2001. Weierstrass K. Mathematical Werke, Bd. 2: Berlin, 1897. Cação I, Malonek HR. 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| References_xml | – reference: Falcão MI, Malonek HR. On paravector valued homogeneous monogenic polynomials with binomial expansion. Advances in Applied Clifford Algebras 2012; 22(3):789-801. – reference: Gürlebeck K, Malonek HR. A hypercomplex derivative of monogenic functions in Rn+1 and its applications. Complex Variables Theory Applications 1999; 39: 199-228. – reference: Appell P. Sur une classe de pôlynomes. Annals scientifiques de l'École Normal Supérieure 21 série 1880; 9(2):119-144. – reference: Cação I, Falcão MI, Malonek HR. Laguerre derivative and monogenic Laguerre polynomials: an operational approach. Mathematical and Computer Modelling 2011; 53(5-6):1084-1094. – reference: Bock S, Gürlebeck K. On a generalized Appell system and monogenic power series. Mathematical Methods In The Applied Sciences 2010; 33(4):394-411. – reference: Cruz C, Falcão MI, Malonek HR. On pseudo-complex bases for monogenic polynomials. AIP Conference Proceedings 2012; 1493: 350-355. – reference: Riordan J. Combinatorial Identities, John Wiley & Sons Inc.: New York, 1968. – reference: Moisil GrC, Théodoresco N. Fonctions holomorphes dans l'espace. Mathematica 1931; 5: 142-159. – reference: Fueter R. Analytische Funktionen einer Quaternionenvariablen. Commentarii Mathematici Helvetici 1932; 4: 9-20. – reference: Sofo A. Computational Techniques for the Summation of Series, Kluwer: New York, 2003. – reference: Lávička R. Canonical bases for sl(2, C)-modules of spherical monogenics in dimension 3. Archivum Mathematicum 2010; 46: 339-349. – reference: Krantz SG. Function Theory of Several Complex Variables, 2nd ed. American Mathematical Society, Providenc: RI, 2001. – reference: Delanghe R. On regular-analytic functions with values in a Clifford algebra. Mathematische Annalen 1970; 185: 91-111. – reference: Cação I, Malonek HR. On complete sets of hypercomplex Appell polynomials. AIP Conference Proceedings 2008; 1048: 647-650. – reference: Cação I, Falcão MI, Malonek HR. Matrix representations of a special polynomial sequence in arbitrary dimension. Computational Methods and Function Theory 2012; 12(2):371-391. – reference: Fueter R. Funktionentheorie im Hyperkomplexen, Lecture notes written and supplemented by E. Bareiss, Math. Inst. Univ. Zürich. Universität Zürich: Herbstsemester,1948/49, 1949. – reference: Delanghe R, Lávička R, Souček V. The Gelfand-Tsetlin bases for Hodge-de Rham systems in Euclidean spaces. Mathematical Methods In The Applied Sciences 2012; 35(7):745-757. – reference: Cruz C, Falcão MI, Malonek HR. 3D mappings by generalized joukowski transformation. Lecture Notes in Computer Science 2011; 6784: 358-373. – reference: Sofo A. Closed form representation of binomial sums and series. Le Matematiche 1999; LIV: 175-186. – reference: Brackx F, Delanghe R, Sommen F. Clifford Analysis. Pitman 76: London, 1982. – reference: Lehmer HD. Interesting series involving the central binomial coefficient. American Mathematical Monthly 1985; 92: 449-457. – reference: Cação I, Falcão MI, Malonek HR. On generalized hypercomplex Laguerre-type exponentials and applications. Lecture Notes in Computer Science 2011; 6784: 271-286. – reference: Brackx F. Two powerful theorems in Clifford analysis. AIP Conference Proceedings 2010; 1281: 3-7. – reference: Malonek HR. Power series representation for monogenic functions in Rn+1 based on a permutational product. Complex Variables, Theory Applications 1990; 15: 181-191. – reference: Weierstrass K. Mathematical Werke, Bd. 2: Berlin, 1897. – reference: Johnson WP. The curious history of Faà di Bruno's formula. The Mathematical Association of America Monthly 2002; 109(3):217-234. – reference: Prodinger H. Knuth's old sum - a survey. European Association for Theoretical Computer Science Bulletin 1994; 54: 232-245. – reference: Malonek HR, de Almeida R. A note on a generalized Joukowski transformation. Applied Mathematics Letters 2010; 23(10):1174-1178. – reference: Gürlebeck K, Habetha K, Sprössig W. Holomorphic Functions in the Plane and n-Dimensional Space, Birkäuser Verlag: Basel, 2008. – reference: Faà di Bruno CF. Note sur une nouvelle formule, de calcul différentiel. Quarterly Journal of Pure and Applied Math 1857; 1: 359-360. – reference: Falcão MI, Malonek HR. Generalized exponentials through Appell sets in Rn+1 and Bessel functions. AIP Conference Procedings 2007; 936: 738-741. – volume: 39 start-page: 199 year: 1999 end-page: 228 article-title: A hypercomplex derivative of monogenic functions in and its applications publication-title: Complex Variables Theory Applications – start-page: 615 year: 2006 end-page: 619 – volume: 15 start-page: 181 year: 1990 end-page: 191 article-title: Power series representation for monogenic functions in based on a permutational product publication-title: Complex Variables, Theory Applications – year: 1968 – volume: 6784 start-page: 358 year: 2011 end-page: 373 article-title: 3D mappings by generalized joukowski transformation publication-title: Lecture Notes in Computer Science – year: 1897 – year: 2001 – year: 2003 – volume: 185 start-page: 91 year: 1970 end-page: 111 article-title: On regular‐analytic functions with values in a Clifford algebra publication-title: Mathematische Annalen – volume: 22 start-page: 789 issue: 3 year: 2012 end-page: 801 article-title: On paravector valued homogeneous monogenic polynomials with binomial expansion publication-title: Advances in Applied Clifford Algebras – volume: 1493 start-page: 350 year: 2012 end-page: 355 article-title: On pseudo‐complex bases for monogenic polynomials publication-title: AIP Conference Proceedings – volume: 92 start-page: 449 year: 1985 end-page: 457 article-title: Interesting series involving the central binomial coefficient publication-title: American Mathematical Monthly – volume: 4 start-page: 9 year: 1932 end-page: 20 article-title: Analytische Funktionen einer Quaternionenvariablen publication-title: Commentarii Mathematici Helvetici – volume: 53 start-page: 1084 issue: 5‐6 year: 2011 end-page: 1094 article-title: Laguerre derivative and monogenic Laguerre polynomials: an operational approach publication-title: Mathematical and Computer Modelling – volume: 12 start-page: 371 issue: 2 year: 2012 end-page: 391 article-title: Matrix representations of a special polynomial sequence in 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| Title | Monogenic pseudo-complex power functions and their applications |
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