Combinations of cross-products of Bessel functions, associated addition formulas and biorthogonal systems

Malits, P.

doi:10.1016/j.jmaa.2018.11.035

Cited by 2 publications

(4 citation statements)

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“…which follows from the integral (16) of the article [22]. The changes 2x = 1−cos θ and 2t = 1 − cos s convert the above integral into a convolution, and then…”

Section: Theorem 3 the Integral Representationmentioning

confidence: 99%

“…where by virtue of analytic continuation Re µ < 1 2 . Finally, by exploiting formulas for derivatives of Bessel functions, we obtain (22), and then (23).…”

Section: Theorem 3 the Integral Representationmentioning

confidence: 99%

“…Make the changes θ = π − ϑ and u = π − s in (22). Then, by exploiting (5) and analytic continuations of cylindrical functions [1], one can obtain as ν = n − 1/2, n ∈ N, and ξ = iη:…”

Section: Theorem 3 the Integral Representationmentioning

confidence: 99%

“…As ξ = 0 and ν = γ + 1/2, the integral representation (22) turns into the famous Mehler-Dirichlet integral:…”

Section: Theorem 3 the Integral Representationmentioning

confidence: 99%

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Ferrers functions of arbitrary degree and order and related functions

Malits¹

2021

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Numerous novel integral and series representations for Ferrers functions of the first kind (associated Legendre functions on the cut) of arbitrary degree and order, various integral, series and differential relations connecting Ferrers functions of different orders and degrees as well as a uniform asymptotic expansion are derived in this article. Elementary proofs of generating functions for Ferrers functions are given. Addition theorems for P −µ ν (tanh (α + β)) are proved by basing on generation functions for three families of hypergeometric polynomials. Relations for Gegenbauer polynomials and Ferrers associated Legendre functions (associated Legendre polynomials) are obtained as special cases.

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“…which follows from the integral (16) of the article [22]. The changes 2x = 1−cos θ and 2t = 1 − cos s convert the above integral into a convolution, and then…”

Section: Theorem 3 the Integral Representationmentioning

confidence: 99%

“…where by virtue of analytic continuation Re µ < 1 2 . Finally, by exploiting formulas for derivatives of Bessel functions, we obtain (22), and then (23).…”

Section: Theorem 3 the Integral Representationmentioning

confidence: 99%

Section: Theorem 3 the Integral Representationmentioning

confidence: 99%

“…As ξ = 0 and ν = γ + 1/2, the integral representation (22) turns into the famous Mehler-Dirichlet integral:…”

Section: Theorem 3 the Integral Representationmentioning

confidence: 99%

See 2 more Smart Citations