On a polynomial sequence associated with the Bessel operator

Loureiro, Ana; Maroni, P.; Yakubovich, S.

doi:10.1090/S0002-9939-2013-11658-8

On a polynomial sequence associated with the Bessel operator
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by Ana F. Loureiro, P. Maroni and S. Yakubovich PDF

Proc. Amer. Math. Soc. 142 (2014), 467-482 Request permission

Abstract:

By means of the Bessel operator a polynomial sequence is constructed to which several properties are given. Among them are its explicit expression, the connection with the Euler numbers, and its integral representation via the Kontorovich-Lebedev transform. Despite its non-orthogonality (with respect to an $L_{2}$-inner product), it is possible to associate to the canonical element of its dual sequence a positive-definite measure as long as certain stronger constraints are imposed.

References

Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vols. I, II, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1953. Based, in part, on notes left by Harry Bateman. MR 0058756
T. S. Chihara, An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR 0481884
Louis Comtet, Advanced combinatorics, Revised and enlarged edition, D. Reidel Publishing Co., Dordrecht, 1974. The art of finite and infinite expansions. MR 0460128
W. N. Everitt, K. H. Kwon, L. L. Littlejohn, R. Wellman, and G. J. Yoon, Jacobi-Stirling numbers, Jacobi polynomials, and the left-definite analysis of the classical Jacobi differential expression, J. Comput. Appl. Math. 208 (2007), no. 1, 29–56. MR 2347735, DOI 10.1016/j.cam.2006.10.045
Yoann Gelineau and Jiang Zeng, Combinatorial interpretations of the Jacobi-Stirling numbers, Electron. J. Combin. 17 (2010), no. 1, Research Paper 70, 17. MR 2651723
N. N. Lebedev, Sur un formule d’inversion, C. R. (Doklady) Acad. Sci. URSS (N.S.) 52 (1946), 655–658 (French). MR 0021144
Ana F. Loureiro, New results on the Bochner condition about classical orthogonal polynomials, J. Math. Anal. Appl. 364 (2010), no. 2, 307–323. MR 2576185, DOI 10.1016/j.jmaa.2009.12.003
Yudell L. Luke, Mathematical functions and their approximations, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0501762
P. Maroni, Une théorie algébrique des polynômes orthogonaux. Application aux polynômes orthogonaux semi-classiques, Orthogonal polynomials and their applications (Erice, 1990) IMACS Ann. Comput. Appl. Math., vol. 9, Baltzer, Basel, 1991, pp. 95–130 (French). MR 1270222
P. Maroni, Variations around classical orthogonal polynomials. Connected problems, Proceedings of the Seventh Spanish Symposium on Orthogonal Polynomials and Applications (VII SPOA) (Granada, 1991), 1993, pp. 133–155. MR 1246855, DOI 10.1016/0377-0427(93)90319-7
P. Maroni, Semi-classical character and finite-type relations between polynomial sequences, Appl. Numer. Math. 31 (1999), no. 3, 295–330. MR 1711161, DOI 10.1016/S0168-9274(98)00137-8
Pascal Maroni and Manoubi Mejri, Generalized Bernoulli polynomials revisited and some other Appell sequences, Georgian Math. J. 12 (2005), no. 4, 697–716. MR 2197192
A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and series. Vol. 1, Gordon & Breach Science Publishers, New York, 1986. Elementary functions; Translated from the Russian and with a preface by N. M. Queen. MR 874986
Walter Van Assche, Open problems, Proceedings of the Seventh Spanish Symposium on Orthogonal Polynomials and Applications (VII SPOA) (Granada, 1991), 1993, pp. 225–243. MR 1246860, DOI 10.1016/0377-0427(93)90324-5
H. M. Srivastava and Á. Pintér, Remarks on some relationships between the Bernoulli and Euler polynomials, Appl. Math. Lett. 17 (2004), no. 4, 375–380. MR 2045740, DOI 10.1016/S0893-9659(04)90077-8
Semyon B. Yakubovich, A class of polynomials and discrete transformations associated with the Kontorovich-Lebedev operators, Integral Transforms Spec. Funct. 20 (2009), no. 7-8, 551–567. MR 2543765, DOI 10.1080/10652460802648473
Semyon B. Yakubovich, On the least values of $L_p$-norms for the Kontorovich-Lebedev transform and its convolution, J. Approx. Theory 131 (2004), no. 2, 231–242. MR 2106539, DOI 10.1016/j.jat.2004.10.007
Semyon B. Yakubovich, On the Kontorovich-Lebedev transformation, J. Integral Equations Appl. 15 (2003), no. 1, 95–112. MR 2004796, DOI 10.1216/jiea/1181074947
S. B. Yakubovich, Index transforms, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. With a foreword by H. M. Srivastava. MR 1391243, DOI 10.1142/9789812831064
Semen B. Yakubovich and Brian Fisher, On the theory of the Kontorovich-Lebedev transformation on distributions, Proc. Amer. Math. Soc. 122 (1994), no. 3, 773–777. MR 1209431, DOI 10.1090/S0002-9939-1994-1209431-0