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Induction proofs of the relations between certain asymptotic expansions and corresponding generalised hypergeometric series. (English) JFM 64.0337.01
Verf. gibt einen neuen Beweis für die bereits von {\it Orr} (Trans. Cambridge philos. Soc. 17 (1898/99), 171-199, 283-290) und {\it Barnes} (Proc. London math. Soc. (2) 5 (1906), 59-116; F. d. M. 38, 449 (JFM 38.0449.*)) angegebenen asymptotischen Darstellungen hypergeometrischer Reihen höherer Ordnung bzw. von Linearkombinationen solcher Reihen, indem er von den bekannten asymptotischen Darstellungen hypergeometrischer Reihen zweiter Ordnung ausgeht und hypergeometrische Reihen höherer Ordnung durch solche niedrigerer Ordnung ausdrückt. Im Anschluß daran werden die Integrale $$ \int\limits_0^{\infty} e^{-x\lambda} F(\alpha,\beta;\gamma;-\lambda)\,d\lambda,\qquad \int\limits_0^1 e^{ix\xi} F(\alpha,\beta;\gamma;\xi)\,d\xi $$ und einige verwandte Integrale mit Kugelfunktionen untersucht.
Reviewer: Erdélyi, A.; Dr. (Edinburgh)