TOPICS
Search

Le Roy Function

The Le Roy function is the entire function defined, for Re[gamma]>0, by

F_gamma(z)=sum_(n=0)^infty(z^n)/((n!)^gamma).
(1)

For gamma=k, where k>=1 is a positive integer, this can be written as the generalized hypergeometric function

F_k(z)=_0F_(k-1)(;1,...,1_()_(k-1);z).
(2)

In particular,

F_1(z)=e^z
(3)
F_2(z)=I_0(2sqrt(z)),
(4)

where I_0 is the modified Bessel function of the first kind.

Guo et al. (2026) use F_k in a generating function for the main-diagonal sequence of the Erdélyi multivariate Laguerre polynomials.

See also

Erdélyi Multivariate Laguerre Polynomial, Generalized Hypergeometric Function, Mittag-Leffler Function

Explore with Wolfram|Alpha

References

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.Le Roy, É. "Sur les séries divergentes et les fonctions définies par un développement de Taylor." Ann. Fac. Sci. Toulouse Math. Ser. 2 2, 317-430, 1900a. https://doi.org/10.5802/afst.173 and https://doi.org/10.5802/afst.174.Le Roy, É. "Valeurs asymptotiques de certaines séries procédant suivant les puissances entières et positives d'une variable réelle." Bull. Sci. Math. Ser. 2 24, 245-268, 1900b.

Cite this as:

Weisstein, Eric W. "Le Roy Function." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LeRoyFunction.html

Subject classifications