Confluent Hypergeometric Limit Function

$${}_0F_1(;a;z)\equiv \lim_{q\to \infty} {}_1F_1\left({q;a;{z\over q}}\right).$$ (1)

It has a series expansion

$${}_0F_1(;a;z) = \sum_{n=0}^\infty {z^n\over(a)_nn!}$$ (2)

and satisfies

$$z{d^2y\over dz^2}+a{dy\over dz}-y=0.$$ (3)

A Bessel Function of the First Kind can be expressed in terms of this function by

$$J_n(x)={({\textstyle{1\over 2}}x)^n\over n!} {}_0F_1(;n+1;-{\textstyle{1\over 4}}x^2)$$ (4)

(Petkovsek et al. 1996).

See also Confluent Hypergeometric Function, Generalized Hypergeometric Function, Hypergeometric Function

References

Petkovsek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A. K. Peters, p. 38, 1996.