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Kummer's Formulas


Kummer's first formula is

 _2F_1(1/2+m-k,-n;2m+1;1)=(Gamma(2m+1)Gamma(m+1/2+k+n))/(Gamma(m+1/2+k)Gamma(2m+1+n)),
(1)

where _2F_1(a,b;c;z) is the hypergeometric function with m!=-1/2, -1, -3/2, ..., and Gamma(z) is the gamma function. The identity can be written in the more symmetrical form as

 _2F_1(a,b;c;-1)=(Gamma(1/2b+1)Gamma(b-a+1))/(Gamma(b+1)Gamma(1/2b-a+1)),
(2)

where a-b+c=1 and b is a positive integer (Bailey 1935, p. 35; Petkovšek et al. 1996; Koepf 1998, p. 32; Hardy 1999, p. 106). If b is a negative integer, the identity takes the form

 _2F_1(a,b;c;-1)=2cos(1/2pib)(Gamma(|b|)Gamma(b-a+1))/(Gamma(1/2b-a+1))
(3)

(Petkovšek et al. 1996).

Kummer's second formula is

M_(0,m)(z)=z^(m+1/2)e^(-z/2)_1F_1(1/2+m;2m+1;z)
(4)
=z^(m+1/2)[1+sum_(k=1)^(infty)(z^(2k))/(2^(4k)k!(m+1)_k)]
(5)
=4^msqrt(z)I_m(1/2z)Gamma(m+1),
(6)

where M_(0,m)(z) is a Whittaker function, _1F_1(a;b;z) is the confluent hypergeometric function of the first kind, (a)_n is a Pochhammer symbol, I_n(z) is a modified Bessel function of the first kind, and m!=-1/2, -1, -3/2, ....


See also

Confluent Hypergeometric Function of the First Kind, Hypergeometric Function

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References

Bailey, W. N. Generalised Hypergeometric Series. Cambridge, England: Cambridge University Press, 1935.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.Koepf, W. Hypergeometric Summation: An Algorithmic Approach to Summation and Special Function Identities. Braunschweig, Germany: Vieweg, 1998.Petkovšek, M.; Wilf, H. S.; and Zeilberger, D. A=B. Wellesley, MA: A K Peters, pp. 42-43 and 126, 1996. http://www.cis.upenn.edu/~wilf/AeqB.html.

Referenced on Wolfram|Alpha

Kummer's Formulas

Cite this as:

Weisstein, Eric W. "Kummer's Formulas." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/KummersFormulas.html

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