A generalization of Bateman's expansion and finite integrals of Sonine's and Feldheim's type

Giacomo Gigante. "A generalization of Bateman's expansion and finite integrals of Sonine's and Feldheim's type." Colloquium Mathematicae 119.2 (2010): 237-254. <http://eudml.org/doc/284271>.

@article{GiacomoGigante2010,
abstract = {Let $\{A_\{k\}\}_\{k=0\}^\{+∞\}$ be a sequence of arbitrary complex numbers, let α,β > -1, let Pₙα,βn=0+∞$ be the Jacobi polynomials and define the functions $$Hₙ(α,z) = ∑_\{m=n\}^\{+∞\} (A_\{m\}z^\{m\})/(Γ(α+n+m+1)(m-n)!)$, $G(α,β,x,y) = ∑_\{r,s=0\}^\{+∞\} (A_\{r+s\}x^\{r\}y^\{s\})/(Γ(α+r+1)Γ(β+s+1)r!s!)$. Then, for any non-negative integer n, $∫_\{0\}^\{π/2\} G(α, β, x²sin²ϕ, y²cos²ϕ) Pₙ^\{α,β\}(cos²ϕ)sin^\{2α+1\}ϕcos^\{2β+1\}ϕd = 1/2 Hₙ(α+β+1,x²+y²) Pₙ^\{α,β\}((y²-x²)/(y²+x²))$. When $A_\{k\} = (-1/4)^\{k\}$, this formula reduces to Bateman’s expansion for Bessel functions. For particular values of y and n one obtains generalizations of several formulas already known for Bessel functions, like Sonine’s first and second finite integrals and certain Neumann series expansions. Particular choices of $\{A_\{k\}\}_\{k=0\}^\{+∞\}$ allow one to write all these type of formulas for specific special functions, like Gegenbauer, Jacobi and Laguerre polynomials, Jacobi functions, or hypergeometric functions.},
author = {Giacomo Gigante},
journal = {Colloquium Mathematicae},
keywords = {Bateman's expansion; Sonine's finite integral; Jacobi polynomials},
language = {eng},
number = {2},
pages = {237-254},
title = {A generalization of Bateman's expansion and finite integrals of Sonine's and Feldheim's type},
url = {http://eudml.org/doc/284271},
volume = {119},
year = {2010},
}

TY - JOUR
AU - Giacomo Gigante
TI - A generalization of Bateman's expansion and finite integrals of Sonine's and Feldheim's type
JO - Colloquium Mathematicae
PY - 2010
VL - 119
IS - 2
SP - 237
EP - 254
AB - Let ${A_{k}}_{k=0}^{+∞}$ be a sequence of arbitrary complex numbers, let α,β > -1, let Pₙα,βn=0+∞$ be the Jacobi polynomials and define the functions $$Hₙ(α,z) = ∑_{m=n}^{+∞} (A_{m}z^{m})/(Γ(α+n+m+1)(m-n)!)$, $G(α,β,x,y) = ∑_{r,s=0}^{+∞} (A_{r+s}x^{r}y^{s})/(Γ(α+r+1)Γ(β+s+1)r!s!)$. Then, for any non-negative integer n, $∫_{0}^{π/2} G(α, β, x²sin²ϕ, y²cos²ϕ) Pₙ^{α,β}(cos²ϕ)sin^{2α+1}ϕcos^{2β+1}ϕd = 1/2 Hₙ(α+β+1,x²+y²) Pₙ^{α,β}((y²-x²)/(y²+x²))$. When $A_{k} = (-1/4)^{k}$, this formula reduces to Bateman’s expansion for Bessel functions. For particular values of y and n one obtains generalizations of several formulas already known for Bessel functions, like Sonine’s first and second finite integrals and certain Neumann series expansions. Particular choices of ${A_{k}}_{k=0}^{+∞}$ allow one to write all these type of formulas for specific special functions, like Gegenbauer, Jacobi and Laguerre polynomials, Jacobi functions, or hypergeometric functions.
LA - eng
KW - Bateman's expansion; Sonine's finite integral; Jacobi polynomials
UR - http://eudml.org/doc/284271
ER -