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Two Proofs of the 6Ψ6 Summation Theorem

Published online by Cambridge University Press:  20 January 2009

L. J. Slater  and
A. Lakin
L. J. Slater
Affiliation:
University Mathematical Laboratory, Cambridge.
A. Lakin
Affiliation:
College of Technology, Manchester.
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The 6Ψ6 summation theorem was first proved by Bailey1, who deduced it indirectly from a transformation of a well-poised 8Φ7 series into two 4Φ3 series. No direct proof of the theorem has been published, and, since it has interesting applications in the proofs of various identities which occur in combinatory analysis, for example the A series of Rogers2 and some elegant identities due to Ramanujan3, we give two new proofs of the theorem in this paper.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 9 , Issue 3 , July 1956 , pp. 116 - 121
Copyright
Copyright © Edinburgh Mathematical Society 1956

References

REFERENCES

[1]Bailey, W. N., “Series of hypergeometric type which are infinite in both directions”, Quart. J. of Math. (Oxford), 7 (1936), 105–115.CrossRefGoogle Scholar
[2]Bailey, W. N., Generalised Hypergeometric Series (Cambridge, 1935).Google Scholar
[3]Bailey, W. N., “A note on two of Ramanujan's formulae”, Quart. J. of Math. (Oxford) (2), 3 (1952), 29–31.CrossRefGoogle Scholar
[4]Burchnall, J. L. and Lakin, A., “The theorems of Saalschiitz and Dougall”, Quart. J. of Math. (Oxford) (2), 1 (1950), 161–164.CrossRefGoogle Scholar
[5]Slater, L. J., “Further identities of the Rogers-Ramanujan type”, Proc. London Math. Soc. (2), 54 (1951), 147–167.Google Scholar