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Abstrakty
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.
$Hₙ(α,z) = ∑_{m=n}^{+∞} (A_{m}z^{m})/(Γ(α+n+m+1)(m-n)!)$,
$G(α,β,x,y) = ∑_{r,s=0}^{+∞} (A_{r+s}x^{r}y^{s})/(Γ(α+r+1)Γ(β+s+1)r!s!)$.
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.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
237-254
Opis fizyczny
Daty
wydano
2010
Twórcy
autor
- Dipartimento di Ingegneria dell'Informazione e Metodi Matematici, Università di Bergamo, Viale Marconi 5, 24044 Dalmine (BG), Italy
Bibliografia
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Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_4064-cm119-2-6