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Riemann P-Series

The solutions to the Riemann P-differential equation are known as the Riemann P-series, or sometimes the Riemann P-function, given by

u(z)=P{a b c; alpha beta gamma; alpha^' beta^' gamma^';z}.
(1)

Solutions are given in terms of the hypergeometric function by

u_1=((z-a)/(z-b))^alpha((z-c)/(z-b))^gamma_2F_1(alpha+beta+gamma,alpha+beta^'+gamma;1+alpha-alpha^';lambda)
(2)
u_2=((z-a)/(z-b))^(alpha^')((z-c)/(z-b))^gamma_2F_1(alpha^'+beta+gamma,alpha^'+beta^'+gamma;1+alpha^'-alpha;lambda)
(3)
u_3=((z-a)/(z-b))^alpha((z-c)/(z-b))^(gamma^')_2F_1(alpha+beta+gamma^',alpha+beta^'+gamma^';1+alpha-alpha^';lambda)
(4)
u_4=((z-a)/(z-b))^(alpha^')((z-c)/(z-b))^(gamma^')_2F_1(alpha^'+beta+gamma^',alpha^'+beta^'+gamma^';1+alpha^'-alpha;lambda),
(5)

where

lambda=((z-a)(c-b))/((z-b)(c-a)).
(6)

See also

Riemann P-Differential Equation

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Riemann's Differential Equation." §15.6 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 564-565, 1972.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 541-543, 1953.Riemann, B. Abh. d. Ges. d. Wiss. zu Göttingen 7, 1857. Reprinted in Mathematisch Werke, p. 67, 1892.Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 283-284, 1990.Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 414, 1995.

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Riemann P-Series

Cite this as:

Weisstein, Eric W. "Riemann P-Series." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/RiemannP-Series.html

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