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

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Gigante G.

Colloquium Mathematicum

2010

tom 119

nr 2

237-254

Let ${A_{k}}_{k=0}^{+∞}$ be a sequence of arbitrary complex numbers, let α,β > -1, let {Pₙ^{α,β}}_{n=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.

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1 Colloquium Mathematicum

1 Gigante G.

1 2010

Wyniki wyszukiwania

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