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On the Riemann P Equation with Integral Exponent differences

Published online by Cambridge University Press:  20 January 2009

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The differential equation in question is of the second order and has three regular singular points. It is usually denoted by

where a, b, c are the singular points, a, a′ etc. the exponents at those singularities. The solution when no one of the numbers α – α′, β – β′, γ – γ′ is an integer or zero is well known; all types of solution are expressible in terms of the hypergeometric functions.

Type
Research Article
Copyright
Copyright © Edinburgh Mathematical Society 1924

References

* (a) Gesammelte Werke VI, 184–202.

(b) Journal für Math. 56 (1859). 149165.Google Scholar

* Quart. J. of Math. 39 (1908) 107.Google Scholar

The notation is best reserved for the functions of §4 which, for λ=0, reduce to associated Legendre Functions.

* When λ = 0, these functions become the associated Legendre functions.