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Riemann P-Differential Equation

The differential equation

(d^2u)/(dz^2)+[(1-alpha-alpha^')/(z-a)+(1-beta-beta^')/(z-b)+(1-gamma-gamma^')/(z-c)](du)/(dz)+[(alphaalpha^'(a-b)(a-c))/(z-a)+(betabeta^'(b-c)(b-a))/(z-b)+(gammagamma^'(c-a)(c-b))/(z-c)]u/((z-a)(z-b)(z-c))=0,

where

alpha+alpha^'+beta+beta^'+gamma+gamma^'=1,

first obtained in the form by Papperitz (1885; Barnes 1908). Solutions are Riemann P-series (Abramowitz and Stegun 1972, pp. 564-565). Zwillinger (1995, p. 414) confusingly calls this equation the "hypergeometric equation."

See also

Heun's Differential Equation, Riemann P-Series

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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.Barnes, E. W. "A New Development in the Theory of the Hypergeometric Functions." Proc. London Math. Soc. 6, 141-177, 1908.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 541-543, 1953.Papperitz. Math. Ann. 25, 213, 1885.Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, 1995.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 126, 1997.

Referenced on Wolfram|Alpha

Riemann P-Differential Equation

Cite this as:

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

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