Addition Theorems for the Functions of the Paraboloid of Revolution
Addition Theorems for the Functions of the Paraboloid of Revolution
Harry Hochstadt
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T, . MS |(-p I, [(^) -\„, (. -, „] - (1**)'^ h [(l**)""^-!^'. *'] from which one derives G (P, t) - GJP', t) + ^ (m. L)G^^^(P, t) * t(in) ^ G^^^ (P, t) . (m-l)G^_^(p', t) - (l*t)4G^_^(p', t)] . One can now state Theorem $ : Under the infinitesimal rotation x' " X + zdT y» - y z' ■ z - xdT one has the fonnula ny) - -Qn^^') * -? [ir^-l*n)Ql*\? ) - (n. I)n^i(p') (m+n-l)n;"^(P') * ( n. L)ll-i-(p')J. - 22 - Acknowledgement The author wishes to express his thanks to Prof. W. Magnus for suggesting the... problem considered here and for his help and interest during the course of the investigation. References [l] J, Meixner and F. W. Schfifke - Kathieusche Funktionen und SpbSroid- funktionenj Springer, 195U» [2] A. Erdelyi, et al. - Higher Transcendental Functions; Vol, 1 and 2, McGraw Hill, 1953. [3] H. Buchhola - Die Konfluente Hypergeometrische Funktionj Springer, 1953* [U] W, Magnus and F, Oberhettinger - Formulas and Theorems for the Special Functions of Mathematical Physicsj Chelsea, 19U9» [^ G.
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