Riemann P-Series

The Riemann -series represents solutions of the Riemann P-differential equation using the symbol

(1)

where , , and are distinct finite regular singular points and the characteristic exponents satisfy . The symbol specifies a class of solutions rather than a uniquely normalized function (NIST, §15.11).

For , with consistent local branches of the powers, the solutions can be expressed in terms of the hypergeometric function as

			(2)
			(3)
			(4)
			(5)

where and denotes the integers. The functions and are linearly independent near , so the general local solution is . The expressions and give alternate forms, rather than two additional independent solutions. Integer exponent differences can require limiting combinations and logarithmic solutions (NIST, §§15.10-15.11).