TOPICS
Search

Hille-Hardy Formula

The Hille-Hardy formula is the Poisson kernel for the associated Laguerre polynomials. For alpha>-1, x,y>0, and 0<z<1, it is

sum_(n=0)^infty(n!)/((alpha+1)_n)L_n^((alpha))(x)L_n^((alpha))(y)z^n=(Gamma(alpha+1)(xyz)^(-alpha/2))/(1-z)exp(-((x+y)z)/(1-z))I_alpha((2sqrt(xyz))/(1-z)),

where I_alpha is the modified Bessel function of the first kind, Gamma is the gamma function, and (alpha+1)_n is the Pochhammer symbol. Erdélyi proved a multivariate extension for the Erdélyi multivariate Laguerre polynomials (Erdélyi 1937, Guo et al. 2026).

See also

Associated Laguerre Polynomial, Erdélyi Multivariate Laguerre Polynomial, Modified Bessel Function of the First Kind, Pochhammer Symbol, Poisson Kernel

Explore with Wolfram|Alpha

References

Erdélyi, A. "Beitrag zur Theorie der konfluenten hypergeometrischen Funktionen von mehreren Veränderlichen." Sitzungsber. Akad. Wiss. Wien, Math.-Naturw. Kl., Abt. IIa 146, 431-467, 1937.Guo, L.-J.; Luo, M.-J.; Raina, R. K.; and Wang, J.-J. "Multivariate Laguerre Polynomials: New Results and Insights." 30 Jul 2026. https://arxiv.org/abs/2604.18629.NIST Digital Library of Mathematical Functions. "Hille-Hardy Formula." §18.18(vii). https://dlmf.nist.gov/18.18.E27.

Cite this as:

Weisstein, Eric W. "Hille-Hardy Formula." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Hille-HardyFormula.html

Subject classifications