Integer composition, connection Appell constants and Bell polynomials
DOI:
https://doi.org/10.33993/jnaat502-1245Keywords:
Appell polynomials, generating functions, connection coefficients, integer composition, connection problems, Bell polynomialsAbstract
We introduce an explicit form of the connection coefficients for Appell polynomial sequences via Toeplitz-Hessenberg matrix determinants.
Generalizing, we give an explicit form of the connection coefficients for arbitrary polynomial sequences and constitute the combinatorial meaning of both constants in terms of integer composition.
Downloads
References
L. Aceto, H.R. Malonek and Gr. Tomaz,A unified matrix approach to the representation of Appell polynomials, Integral Transforms. Spec. Funct., 26, (2015), no.6, pp. 426–441. https://doi.org/10.1080/10652469.2015.1013035
J.A. Adell, andA. Lekuona, Binomial convolution and transformations of Appell polynomials, J. Math.Annal. Appl., 456, (2017), no.1, pp. 16–33, https://doi.org/10.1016/j.jmaa.2017.06.077
P. Appell, On one class of polynomials, Ann. Sci. Ec. Norm. Super., 9, (1880), 2e serie, pp. 119–144.
G. E. Andrews,The Theory of Partitions, Encyclopedia of Mathematics and Its Applications, Vol. 2, Addison-Wesley, 1976.
H. Belbachir, S. Haj Brahim and M.Rachidi, On another approach for a family of Appell polynomials, Filomat, 9, (2018), pp. 4155–4164. http://doi.org/10.2298/FIL1812155B
E.T. Bell, Partition Polynomials, Ann. Math., Second Series, 29, (1927-1928), no.1/4, pp. 38–46, https://doi.org/10.2307/1967979
N. Bonneux, Z. Hamaker , J. Stembridge and M. Stevensa, Wronskian Appell polynomials and symmetric functions Adv. Appl. Math., (2019) , v. 111, 101932. https://doi.org/10.1016/j.aam.2019.101932
F. Brioschi , Sulle funzioni Bernoulliane ed Euleriane, Ann. Mat. Pura Appl., (1858), pp. 260-263, https://doi.org/10.1007/BF03197335
S. A. Carillo and M.Hurtado , J. Stembridge and M. Stevensa, Appell and Sheffer sequences; on their characterizations through functionals and exmples, Comptes Rendus Math., (2021) , no. 2, pp. 205–217, https://doi.org/10.5802/crmath.172
F. A. Costabile and E.Longo, A determinantal approach to Appell polynomials, J. Comput. Appl. Math., 234, (2010) , no. 2, pp. 1528–1542, https://doi.org/10.1016/j.cam.2010.02.033
Y.B. Cheikh, H. Chaggara, Connection problems via lowering operators, J. Comput. Appl. Math., 178, (2005), no. (1-2), pp. 45-61. http://doi.org/10.1016/j.cam.2004.02.02
G. Dattoli, Hermite-Bessel and Laguerre-Bessel functions: a by-product of the monomiality principle in Advanced Special Functions and Applications, Proceedings of the Melfi School on Advanced Topicsin Mathematics and Physics, J. Comput. Appl. Math., Aracne Editrici, Rome, (2005), pp. 147-164. http://doi.org/10.1016/j.cam.2004.02.024
E. Godoy, A. Ronveaux, A. Zarzo and I. Area, Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: Continuous case. J. Comput. Appl. Math., (1997), no. 84, pp. 257-275, https://doi.org/10.1016/S0377-0427(97)00137-4
W. Koepf and D. Schmersau, Recurrence equations and their classical orthogonal polynomials solutions, Appl. Math. Comput., (2002), no. 128, pp. 303-327, https://doi.org/10.1016/S0096-3003(01)00078-9
S. Lewanowicz ,The Hypergeometric functions approach to the connection problem for the classical orthogonal polynomials, Inst. of Computer Sci., Univ. of Wroclaw, 2003.
J. C. Lopez, R. Carreno, R. M. Suarez, J. A. Mendoza, Connection Formulae among Special Polynomials, Int. J. Math. Comput. Sci., 10 (2015), no. 1, pp. 39-49.
N. Luno, Connection problems for the generalized hypergeometric Appell polynomials, Proceedings of the International Geometry Center, 13, (2020), no. 2, pp. 1–18. http://doi.org/10.15673/tmgc.v13i2.1733
M. Merca, A note on the determinant of a Toeplitz-Hessenberg matrix, Spec. Matrices, (2013), pp. 10–16. https://eudml.org/doc/267216
H.D. Nguyen and L.G. Cheong, New convolution identities for hypergeometric Bernoulli polynomials, J. Number Theory, 137, (2014), pp. 201-221. https://doi.org/10.1016/j.jnt.2013.11.008
H. Pan and Zh.W .Sun, New identities involving Bernoulli and Euler polynomials, 113, (2006), no. 1, pp. 156–175, https://doi.org/10.1016/j.jcta.2005.07.008
J. Quaintance, Combinatorial Identities for Stirling Numbers: The Unpublished Notes of H. W. Gould., World Scientific Publishing, Singapore, 2015.
S. Roman, G.-C. Rota, The umbral calculus, Adv. Math., 27, (1978), is.2, pp. 95–188, https://doi.org/10.1016/0001-8708(78)90087-7
S. Roman, The umbral calculus, Dover Publ. Inc., New York, 2005.
A. Ronveaux ), Orthogonal polynomials: connection and linearisation coefficients. Proceedings of the International Workshop on Orthogonal Polynomials in Mathematical Physics. Leganes, 24-26 June, 1996.
J. Snchez-Ruiz and J. S. Dehesa, Some connection and linearization problems for the polynomials in and beyond the Askey scheme, J. Comput. Appl. Math., (2001), no. 133, pp. 579-591, http://doi.org/10.1016/S0377-0427(00)00679-8
Published
How to Cite
Issue
Section
License
Copyright (c) 2022 Journal of Numerical Analysis and Approximation Theory
This work is licensed under a Creative Commons Attribution 4.0 International License.
Open Access. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
