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

• Autorzy: Giacomo Gigante
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
• 42C05: Orthogonal functions and polynomials, general theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2010
• Tom: 119
• Numer: 2
• Strony: 237-254

Daty

wydano

2010

Twórcy

autor

Giacomo Gigante

• Dipartimento di Ingegneria dell'Informazione e Metodi Matematici, Università di Bergamo, Viale Marconi 5, 24044 Dalmine (BG), Italy

Identyfikatory

DOI

10.4064/cm119-2-6