Abstract
Our main objective is to establish the so-called connection formula,
which for p n(x) = x n is known as the inversion formula
for the family y k(x), where \(\{p_n(x)\}_{n\in \mathbb {N}_0}\) and \(\{y_n(x)\}_{n\in \mathbb {N}_0}\) are two polynomial systems. If we substitute x by ax in the left hand side of (0.1) and y k by p k, we get the multiplication formula
The coefficients C k(n), I k(n) and D k(n, a) exist and are unique since deg p n = n, deg y k = k and the polynomials {p k(x), k = 0, 1, …, n} or {y k(x), k = 0, 1, …, n} are therefore linearly independent. In this session, we show how to use generating functions or the structure relations to compute the coefficients C k(n), I k(n) and D k(n, a) for classical continuous orthogonal polynomials.
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References
I. Area, E. Godoy, A. Ronveaux, A. Zarzo, Solving connection and linearization problems within the Askey scheme and its q-analogue via inversion formulas. J. Comput. Appl. Math. 136, 152–162 (2001)
R. Askey, G. Gasper, Jacobi polynomial expansions of Jacobi polynomials with nonnegative coefficients. Proc. Camb. Philos. Soc. 70, 243–255 (1971)
R. Askey, G. Gasper, Convolution structures for Laguerre polynomials. J. Anal. Math. 31, 48–68 (1977)
H. Chaggara, W. Koepf, Duplication coefficients via generating functions. Complex Var. Elliptic Equ. 52, 537–549 (2007)
T. Cluzeau, M. van Hoeij, Computing hypergeometric solutions of linear recurrence equations. Appl. Algebra Eng. Commun. Comput. 17, 83–115 (2006)
E.H. Doha, H.M. Ahmed, Recurrences and explicit formulae for the expansion and connection coefficients in series of Bessel polynomials. J. Phys. A 37, 8045–8063 (2004)
J.L. Fields, J. Wimp, Expansions of hypergeometric functions in hypergeometric functions. Math. Comp. 15, 390–395 (1961)
E. Godoy, A. Ronveaux, A. Zarzo, I. Area, Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: continuous case. J. Comput. Appl. Math. 84, 257–275 (1997)
M.E.H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable. Encyclopedia of Mathematics and its Applications, vol. 98 (Cambridge University Press, Cambridge, 2005)
W. Koepf, Hypergeometric Summation—An Algorithmic Approach to Summation and Special Function Identities, 2nd edn. (Springer Universitext, Springer, London, 2014)
W. Koepf, D. Schmersau, Representations of orthogonal polynomials. J. Comput. Appl. Math. 90, 57–94 (1998)
S. Lewanowicz, The hypergeometric functions approach to the connection problem for the classical orthogonal polynomials. Technical Report, Institute of Computer Science, University of Wroclaw (2003)
P. Njionou Sadjang, Moments of classical orthogonal polynomials, Ph.D. Dissertation, Universität Kassel (2013)
M. Petkovšek, Hypergeometric solutions of linear recurrences with polynomial coefficients. J. Symb. Comput. 14, 243–264 (1992)
M. Petkovšek, H. Wilf, D. Zeilberger, A = B (A. K. Peters, Wellesley, 1996)
E.D. Rainville, Special Functions (The Macmillan Company, New York, 1960)
A. Ronveaux, A. Zarzo, E. Godoy, Recurrence relations for connection between two families of orthogonal polynomials. J. Comput. Appl. Math. 62, 67–73 (1995)
J. Sánchez-Ruiz, J.S. Dehesa, Expansions in series of orthogonal hypergeometric polynomials. J. Comput. Appl. Math. 89, 155–170 (1997)
D.D. Tcheutia, On connection, linearization and duplication coefficients of classical orthogonal polynomials, Ph.D. Dissertation, Universität Kassel (2014)
D.D. Tcheutia, M. Foupouagnigni, W. Koepf, P. Njionou Sadjang, Coefficients of multiplication formulas for classical orthogonal polynomials. Ramanujan J. 39, 497–531 (2016)
M. van Hoeij, Finite singularities and hypergeometric solutions of linear recurrence equations. J. Pure Appl. Algebra 139, 109–131 (1999)
A. Zarzo, I. Area, E. Godoy, A. Ronveaux, Results for some inversion problems for classical continuous and discrete orthogonal polynomials. J. Phys. A 30, 35–40 (1997)
Acknowledgements
Many thanks to the organizers of the AIMS–Volkswagen Stiftung Workshop on Introduction to Orthogonal Polynomials and Applications, Douala, October 5–12, 2018.
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Tcheutia, D.D. (2020). Inversion, Multiplication and Connection Formulae of Classical Continuous Orthogonal Polynomials. In: Foupouagnigni, M., Koepf, W. (eds) Orthogonal Polynomials. AIMSVSW 2018. Tutorials, Schools, and Workshops in the Mathematical Sciences . Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-36744-2_5
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