A GENERALIZATION OF BATEMAN'S EXPANSION AND FINITE INTEGRALS OF SONINE'S AND FELDHEIM'S TYPE

作者：Gigante Giacomo^*

来源：Colloquium Mathematicum, 2010, 119(2): 237-254.

DOI：10.4064/cm119-2-6

• Bateman's expansion
• Sonine's finite integral
• Jacobi polynomials

Let {A(k)}(k=0)(+infinity) be a sequence of arbitrary complex numbers, let alpha, beta > -1, let {P(n)(alpha,beta)}(n=0)(+infinity) be the Jacobi polynomials and define the functions
Hn(alpha, z) = Sigma(+infinity)(m=n)Am(zm)/Gamma(alpha + n + m + 1)(m - n)!'
G(alpha, beta, x, y) = Sigma(+infinity)(r,s-0)A(r+s)x(r)y(s)/Gamma(alpha + r + 1)Gamma(beta + s + 1)r!s!(+)
Then, for any non-negative integer n,
integral(pi/2)(0) G(alpha, beta, x(2) sin(2) phi, y(2) cos(2) phi)P(n)(alpha,beta)(cos2 phi) sin 2(alpha+1) phi cos(2 beta+1) phi d phi
= 1/2H(n)(alpha + beta + 1, x(2) + y(2))P(n)(alpha,beta) (y(2) - x(2)/y(2) + x(2))
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)(+infinity) allow one to write all these type of formulas for specific special functions, like Gegenbauer, Jacobi and Laguerre polynomials, Jacobi functions, or hypergeometric functions.

• 出版日期2010