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On the Relation Between Gegenbauer Polynomials and the Ferrers Function of the First Kind

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Abstract

Using the direct relation between the Gegenbauer polynomials \(C_n^\lambda(x)\) and the Ferrers function of the first kind \({\rm{P}}_\nu^\mu(x)\), we compute interrelations between certain Jacobi polynomials, Meixner polynomials, and Ferrers functions of the first and second kind. We then compute Rodrigues-type, standard integral orthogonality and Sobolev orthogonality relations for Ferrers functions of the first and second kinds. In the remainder of the paper using the relation between Gegenbauer polynomials and the Ferrers function of the first kind we derive connection and linearization relations, some definite integral and series expansions, Christoffel-Darboux summation formulas, Poisson kernel and infinite series closure relations (Dirac delta distribution expansions).

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Acknowledgement

Much thanks to Adri Olde Daalhuis and Nico Temme for valuable discussions.

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Authors and Affiliations

  1. Applied and Computational Mathematics Division, National Institute of Standards and Technology, Mission Viejo, CA, 92694, USA

    H. S. Cohl

  2. Dpto. de Física y Matemáticas, Universidad de Alcalá, c.p. 28871, Alcalá de Henares, Spain

    R. S. Costas-Santos

Authors
  1. H. S. Cohl
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  2. R. S. Costas-Santos
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Correspondence to R. S. Costas-Santos.

Additional information

The research of R.S. Costas-Santos was funded by Agencia Estatal de Investigación of Spain, grant number PGC-2018-096504-B-C33.

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Cohl, H.S., Costas-Santos, R.S. On the Relation Between Gegenbauer Polynomials and the Ferrers Function of the First Kind. Anal Math 48, 695–716 (2022). https://doi.org/10.1007/s10476-022-0123-0

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  • DOI: https://doi.org/10.1007/s10476-022-0123-0

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