By means of Abel’s method on summation by parts, some two term recurrence relations on very wellpoised $_6\psi_6$-series are established. Their iteration yields a $_6\psi_6$-series transformation with an extra natural number parameter. Evaluating the limiting series via Jacobi’s triple product identity, we are led surprisingly to the celebrated bilateral $_6\psi_6$-series identity discovered by Bailey (1936). Then we shall further generalize it to a very well-poised 10ψ10-series identity, which contains Shukla’s formula (1959) as special case. Finally, the Abel’s method on summation by parts will be employed again to investigate the bibasic hypergeometric series summation, which may be considered as an extension of a “split-poised” transformation on terminating $_{10}\phi_9$-series due to Gasper (1989).

Bailey's Very Well-Poised 6psi6-Series Identity

CHU, Wenchang
2006-01-01

Abstract

By means of Abel’s method on summation by parts, some two term recurrence relations on very wellpoised $_6\psi_6$-series are established. Their iteration yields a $_6\psi_6$-series transformation with an extra natural number parameter. Evaluating the limiting series via Jacobi’s triple product identity, we are led surprisingly to the celebrated bilateral $_6\psi_6$-series identity discovered by Bailey (1936). Then we shall further generalize it to a very well-poised 10ψ10-series identity, which contains Shukla’s formula (1959) as special case. Finally, the Abel’s method on summation by parts will be employed again to investigate the bibasic hypergeometric series summation, which may be considered as an extension of a “split-poised” transformation on terminating $_{10}\phi_9$-series due to Gasper (1989).
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11587/103590
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