Riemann P-Series

The solutions to the Riemann P-Differential Equation

$$z=P\left\{{\matrix{ a & b & c\cr \alpha & \beta & \gamma\cr \alpha' & \beta' & \gamma'\cr}\,;\, z}\right\}.$$

Solutions are given in terms of the Hypergeometric Function by

$$u_1 = \left({z-a\over z-b}\right)^\alpha \left({z-c\over z-b}\right)^\g... ...\, {}_2F_1(\alpha+\beta+\gamma,\alpha+\beta'+\gamma; 1+\alpha-\alpha'; \lambda)$$

$$u_2 = \left({z-a\over z-b}\right)^{\alpha'} \left({z-c\over z-b}\right)... ... {}_2F_1(\alpha'+\beta+\gamma,\alpha'+\beta'+\gamma; 1+\alpha'-\alpha; \lambda)$$

$$u_3 = \left({z-a\over z-b}\right)^\alpha \left({z-c\over z-b}\right)^{\... ... {}_2F_1(\alpha+\beta+\gamma',\alpha+\beta'+\gamma'; 1+\alpha-\alpha'; \lambda)$$

$$u_4 = \left({z-a\over z-b}\right)^{\alpha'} \left({z-c\over z-b}\right)... ...2F_1(\alpha'+\beta+\gamma', \alpha'+\beta'+\gamma'; 1+\alpha'-\alpha; \lambda),$$

where

$$\lambda\equiv{(z-a)(c-b)\over (z-b)(c-a)}.$$

References

Abramowitz, M. and Stegun, C. 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.

Whittaker, E. T. and Watson, G. N. A Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University Press, pp. 283-284, 1990.