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Euler's Hypergeometric Transformations

_2F_1(a,b;c;z)=int_0^1(t^(b-1)(1-t)^(c-b-1))/((1-tz)^a)dt,
(1)

where _2F_1(a,b;c;z) is a hypergeometric function. The solution can be written using the Euler's transformations

t->t
(2)
t->1-t
(3)
t->(1-z-tz)^(-1)
(4)
t->(1-t)/(1-tz)
(5)

in the equivalent forms

_2F_1(a,b;c;z)=(1-z)^(-a)_2F_1(a,c-b;c;z/(z-1))
(6)
=(1-z)^(-b)_2F_1(c-a,b;c;z/(z-1))
(7)
=(1-z)^(c-a-b)_2F_1(c-a,c-b;c;z).
(8)

Equation (7) gives Euler's convergence improvement transform of the series _2F_1(a,b;c;-1) (Abramowitz and Stegun 1972, p. 555).

See also

Euler Transform, Hypergeometric Function

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References

Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, 1972.Euler, L. Nova Acta Acad. Petropol. 7, p. 58, 1778.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 585-591, 1953.

Referenced on Wolfram|Alpha

Euler's Hypergeometric Transformations

Cite this as:

Weisstein, Eric W. "Euler's Hypergeometric Transformations." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/EulersHypergeometricTransformations.html

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