On Quadratic Transformations of Basic Series

Al‐Salam, W. A.; Verma, Arun

doi:10.1137/0515032

Cited by 26 publications

(30 citation statements)

References 10 publications

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“…Eleven years later, Al-Salam and Verma [3] used the fact that the «th indifference of a polynomial of degree less than « is equal to zero to derive the bibasic summation formula ( _£\ A (-i)W;«),., (°-7)=s They also observed that (1.7) is equivalent to the fact that the triangular matrix 77 = (hnj), where nn h (-^rJ(aqp";q)n_x(l-ap]q]) nj (p-,P)n-j(aqp ;q)j is inverse to the triangular matrix G= (gjn), where (19) g. JapHqn\q)!-»pW.…”

Section: Introductionmentioning

confidence: 99%

Summation, transformation, and expansion formulas for bibasic series

Gasper¹

1989

Trans. Amer. Math. Soc.

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Section: Introductionmentioning

confidence: 99%

Summation, transformation, and expansion formulas for bibasic series

Gasper¹

1989

Trans. Amer. Math. Soc.

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“…There is of course ample precedent for base-changing transformations; see, e.g., [1,13,20,28,29,30,52,53,56].…”

Section: Positivity Preserving Q-binomial Transformationsmentioning

confidence: 99%

Positivity preserving transformations for 𝑞-binomial coefficients

Berkovich

Warnaar

2004

Trans. Amer. Math. Soc.

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Abstract. Several new transformations for q-binomial coefficients are found, which have the special feature that the kernel is a polynomial with nonnegative coefficients. By studying the group-like properties of these positivity preserving transformations, as well as their connection with the Bailey lemma, many new summation and transformation formulas for basic hypergeometric series are found. The new q-binomial transformations are also applied to obtain multisum Rogers-Ramanujan identities, to find new representations for the Rogers-Szegö polynomials, and to make some progress on Bressoud's generalized Borwein conjecture. For the original Borwein conjecture we formulate a refinement based on new triple sum representations of the Borwein polynomials.

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“…However, the evaluations that we used were the standard ones-Vandermonde's theorem, Saalschütz' theorem, and the very well-poised 6<ï>5. It is quite possible that other evaluations could give other transformations [3,Equation (3.3)]. Verma and Jain have given several bibasic evaluations which might work [28].…”

Section: =0mentioning

confidence: 99%

“…It can be used to give a systematic approach to ^-analogs of quadratic transformations. There is a scattering of such results in the literature [3,16,27,29,33]. However, the techniques are verifications by manipulating sums.…”

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confidence: 99%