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arxiv: 2411.03571 · v4 · submitted 2024-11-06 · 🧮 math.CA

Product formulas for basic hypergeometric series by evaluations of Askey--Wilson polynomials

Howard Cohl , Michael Schlosser This is my paper

Pith reviewed 2026-05-23 18:05 UTC · model grok-4.3

classification 🧮 math.CA
keywords Askey-Wilson polynomialsbasic hypergeometric seriesgenerating functionssummation formulasproduct formulasq-quadratic special valuesterminating summations
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The pith

A generalization of the Ismail-Wilson generating function for Askey-Wilson polynomials yields product formulas for basic hypergeometric series.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper generalizes an existing generating function for Askey-Wilson polynomials by adding one extra parameter. A special case of this generalization provides a closed-form expression for a quadruple sum of basic hypergeometric terms. The work also introduces new terminating summation formulas for balanced 4 phi 3 series that correspond to q-quadratic special values of the Askey-Wilson polynomials, along with 2-balanced and 3-balanced variants. These results are then combined with existing summations to produce new product transformations and integral representations for nonterminating basic hypergeometric series.

Core claim

The authors derive a generalization of the Ismail-Wilson generating function for Askey-Wilson polynomials that incorporates an additional parameter. Special cases of this formula give a closed summation for a quadruple basic hypergeometric sum and new terminating balanced 4 phi 3 summations that evaluate to q-quadratic special values for Askey-Wilson polynomials. Similar new summations that are 2-balanced and 3-balanced are also given. Combining these with the original generating function produces new product transformations for nonterminating series along with integral representations, and further identities arise from Cayley-Orr type expansion formulas.

What carries the argument

The generalized generating function with an extra parameter, expressed as a product of q-Gauss functions, used to derive summation formulas via evaluations of Askey-Wilson polynomials.

Load-bearing premise

The new identities are assumed to hold for generic complex parameters satisfying the standard convergence and non-vanishing conditions of the nonterminating and terminating basic hypergeometric series involved.

What would settle it

Numerical evaluation of both sides of one of the new summation formulas for a concrete choice of parameters where all series converge, to check whether equality holds.

read the original abstract

Ismail and Wilson derived a generating function for Askey--Wilson polynomials which is given by a product of $q$-Gauss (Heine) nonterminating basic hypergeometric functions. We provide a generalization of that generating function which contains an extra parameter. A special case gives a closed form summation formula for a quadruple basic hypergeometric sum. We further present new terminating balanced ${}_4\phi_3$ summations that give rise to $q$-quadratic special values for Askey--Wilson polynomials. We also similarly present new terminating 2-balanced and 3-balanced ${}_4\phi_3$ summations. Using the Ismail--Wilson generating function combined with explicit summations for terminating balanced basic hypergeometric $_4\phi_3$ series, we compute new basic hypergeometric product transformations for nonterminating basic hypergeometric series and provide corresponding integral representations. Further new identities are obtained by applying Cayley--Orr type expansion formulas.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 4 minor

Summary. The paper generalizes the Ismail-Wilson generating function for Askey-Wilson polynomials to include an extra parameter, yielding a product of q-Gauss functions. A special case produces a closed-form summation formula for a quadruple basic hypergeometric sum. It derives new terminating balanced _4 phi_3 summations (as well as 2-balanced and 3-balanced variants) that give q-quadratic evaluations of Askey-Wilson polynomials. Combining the generating function with known summations, the authors obtain new product transformations for nonterminating series, associated integral representations, and further identities via Cayley-Orr expansions.

Significance. If the algebraic derivations hold under the stated generic-parameter assumptions, the work supplies concrete new identities in basic hypergeometric series that extend the Ismail-Wilson formula and provide explicit q-quadratic special values. The quadruple-sum closed form and the balanced _4 phi_3 summations are the most load-bearing contributions; they rest on standard manipulations rather than new analytic machinery and could serve as tools for further q-series research.

minor comments (4)
  1. §2, after Eq. (2.3): the statement of the generalized generating function would benefit from an explicit display of the extra parameter's range relative to the original Ismail-Wilson conditions to avoid any ambiguity in the nonterminating case.
  2. Theorem 3.2 and Corollary 3.3: the transition from the quadruple sum to the product formula is presented without an intermediate step showing how the extra parameter is specialized; inserting one line of algebra would improve readability.
  3. §4, the 2-balanced and 3-balanced _4 phi_3 identities: the convergence conditions are stated only by reference to the generic case; a short sentence listing the precise non-vanishing denominator requirements would prevent reader uncertainty.
  4. The integral representations in §5 are given without a brief remark on the contour or the q-integral definition used; adding this would make the section self-contained.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper derives new product formulas and summation identities for basic hypergeometric series by starting from the externally cited Ismail-Wilson generating function (distinct authors) together with known terminating balanced 4phi3 summations. These are combined via algebraic manipulations and Cayley-Orr expansions to produce the claimed generalizations and q-quadratic evaluations. No derivation step reduces a claimed result to its own inputs by definition, renames a fitted quantity as a prediction, or relies on a load-bearing self-citation chain; the central claims remain independent of the paper's own outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on the standard analytic continuation and convergence theory of basic hypergeometric series; no new free parameters, ad-hoc axioms, or invented entities are introduced beyond the extra parameter in the generating function itself.

axioms (1)
  • domain assumption Standard convergence and non-vanishing conditions for nonterminating and terminating basic hypergeometric series
    Invoked implicitly to guarantee that the generating function and summation formulas are well-defined.

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Works this paper leans on

23 extracted references · 23 canonical work pages

  1. [1]

    W. N. Bailey. Products of Generalized Hypergeometric Series. Proceedings of the London Mathe- matical Society. Second Series, 28(4):242–254, 1928

    work page 1928

  2. [2]

    W. N. Bailey. A note on certain q-identities. The Quarterly Journal of Mathematics. Oxford Series , 12:173–175, 1941

    work page 1941

  3. [3]

    W. N. Bailey. Generalized hypergeometric series. Cambridge Tracts in Mathematics and Mathemat- ical Physics, No. 32. Stechert-Hafner, Inc., New York, 1964

    work page 1964

  4. [4]

    Berkovich and S

    A. Berkovich and S. O. Warnaar. Positivity preserving transformations for q-binomial coefficients. Transactions of the American Mathematical Society , 357(6):2291–2351, 2005

    work page 2005

  5. [5]

    L. Carlitz. Some formulas of F. H. Jackson. Monatshefte f¨ ur Mathematik, 73:193–198, 1969

    work page 1969

  6. [6]

    T. Clausen. ¨Uber die F¨ alle, wenn die Reihe von der Form y = 1 + α 1 · β γ x + α · α + 1 1 · 2 · β · β + 1 γ · γ + 1 x2 + etc. ein Quadrat von der Form z = 1 + α′ 1 · β′ γ′ · δ′ ε′ x + α′ · α′ + 1 1 · 2 · β′ · β′ + 1 γ′ · γ′ + 1 · δ′δ′ + 1 ε′ε′ + 1x2 + etc. hat. Journal f¨ ur die Reine und Angewandte Mathematik, 3:89–91, 1828

    work page

  7. [7]

    H. S. Cohl and R. S. Costas-Santos. Symmetry of terminating basic hypergeometric representations of the Askey-Wilson polynomials. Journal of Mathematical Analysis and Applications, 517(1):126583, 2023

    work page 2023

  8. [8]

    H. S. Cohl and R. S. Costas-Santos. Utility of integral representations for basic hypergeometric functions and orthogonal polynomials. The Ramanujan Journal, Special Issues in Memory of Richard Askey, 61:649–674, May 2023. Product formulas for basic hypergeometric series 21

    work page 2023

  9. [9]

    https://dlmf.nist.gov/, Release 1.2.3 of 2024- 12-15

    NIST Digital Library of Mathematical Functions . https://dlmf.nist.gov/, Release 1.2.3 of 2024- 12-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V. Saunders, H. S. Cohl, and M. A. McClain, eds

    work page 2024

  10. [10]

    Gasper and M

    G. Gasper and M. Rahman. Basic hypergeometric series, volume 96 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, second edition, 2004. With a foreword by Richard Askey

    work page 2004

  11. [11]

    M. E. H. Ismail. Classical and Quantum Orthogonal Polynomials in One Variable , volume 98 of Encyclopedia of Mathematics and its Applications . Cambridge University Press, Cambridge, 2005. With two chapters by Walter Van Assche

    work page 2005

  12. [12]

    M. E. H. Ismail and J. A. Wilson. Asymptotic and generating relations for the q-Jacobi and 4φ3 polynomials. Journal of Approximation Theory , 36(1):43–54, 1982

    work page 1982

  13. [13]

    F. H. Jackson. The qθ equations whose solutions are products of solutions of qθ equations of lower order. The Quarterly Journal of Mathematics. Oxford Series , 11:1–17, 1940

    work page 1940

  14. [14]

    F. H. Jackson. Certain q-identities. The Quarterly Journal of Mathematics. Oxford Series , 12:167– 172, 1941

    work page 1941

  15. [15]

    Koekoek, P

    R. Koekoek, P. A. Lesky, and R. F. Swarttouw. Hypergeometric orthogonal polynomials and their q-analogues. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2010. With a foreword by Tom H. Koornwinder

    work page 2010

  16. [16]

    Hypergeometric orthogonal polynomi- als and their q-analogues

    T. H. Koornwinder. Additions to the formula lists in “Hypergeometric orthogonal polynomi- als and their q-analogues” by Koekoek, Lesky and Swarttouw. arXiv:1401.0815v2, see also https://staff.fnwi.uva.nl/t.h.koornwinder/art/informal/KLSadd.pdf, 2024

    work page arXiv 2024

  17. [17]

    B. G. Nassrallah. Some quadratic transformations and projection formulas for basic hypergeometric series. PhD thesis, Carleton University, Ottowa, Canada, 1982. xiv+190 pages

    work page 1982

  18. [18]

    W. M. Orr. Theorems relating to the product of two hypergeometric series. Transactions of the Cambridge Philosophical Society, 17:1–15, 1899

    work page

  19. [19]

    Paule and M

    P. Paule and M. Schorn. A Mathematica version of Zeilberger’s algorithm for proving binomial co- efficient identities. Journal of Symbolic Computation , 20(5-6):673–698, 1995. Symbolic computation in combinatorics ∆1 (Ithaca, NY, 1993)

    work page 1995

  20. [20]

    Petkovˇ sek, H

    M. Petkovˇ sek, H. S. Wilf, and D. Zeilberger. A = B. A K Peters Ltd., Wellesley, MA, 1996. With a foreword by Donald E. Knuth, With a separately available computer disk

    work page 1996

  21. [21]

    M. J. Schlosser. q-Analogues of Two Product Formulas of Hypergeometric Functions by Bailey. In Frontiers in Orthogonal Polynomials and q-Series, chapter 23, pages 445–449. World Scientific Publishing, Hackensack, NJ, 2018. Zuhair Nashed and Xin Li, editors

    work page 2018

  22. [22]

    H. M. Srivastava. Some formulas of Srinivasa Ramanujan involving products of hypergeometric functions. Indian Journal of Mathematics , 29(1):91–100, 1987

    work page 1987

  23. [23]

    H. M. Srivastava and V. K. Jain. q-series identities and reducibility of basic double hypergeometric functions. Canadian Journal of Mathematics. Journal Canadien de Math´ ematiques, 38(1):215–231, 1986

    work page 1986