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Confluent Hypergeometric Limit Function

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_0F_1(;a;z)=lim_(q->infty)_1F_1(q;a;z/q).
(1)

It has a series expansion

_0F_1(;a;z)=sum_(n=0)^infty(z^n)/((a)_nn!)
(2)

and satisfies

z(d^2y)/(dz^2)+a(dy)/(dz)-y=0.
(3)

It is implemented in the Wolfram Language as Hypergeometric0F1[b, z].

A Bessel function of the first kind can be expressed in terms of this function by

J_n(x)=((1/2x)^n)/(n!)_0F_1(;n+1;-1/4x^2)
(4)

(Petkovšek et al. 1996).

See also

Confluent Hypergeometric Function of the First Kind, Generalized Hypergeometric Function, Hypergeometric Function

Related Wolfram sites

http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric0F1/, http://functions.wolfram.com/HypergeometricFunctions/Hypergeometric0F1Regularized/

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References

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

Referenced on Wolfram|Alpha

Confluent Hypergeometric Limit Function

Cite this as:

Weisstein, Eric W. "Confluent Hypergeometric Limit Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/ConfluentHypergeometricLimitFunction.html

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