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arxiv: 2605.06248 · v1 · submitted 2026-05-07 · 🧮 math.CO

Double-sum Rogers-Ramanujan type identities

Duanyu Chen , Xiangxin Liu , Lisa Hui Sun This is my paper

Pith reviewed 2026-05-08 08:14 UTC · model grok-4.3

classification 🧮 math.CO
keywords Rogers-Ramanujan identitiesdouble-sum identitiesq-Hermite polynomialsChebyshev polynomialsq-orthogonal polynomialsBailey pairspartition identities
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The pith

Expansion of Chebyshev polynomials in q-Hermite polynomials yields double-sum Rogers-Ramanujan identities.

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

The paper aims to derive a series of Rogers-Ramanujan type identities that feature sums over two variables rather than one. It proceeds by inserting the known expansion of Chebyshev polynomials as linear combinations of q-Hermite polynomials into the orthogonality relations satisfied by the q-Hermite family. The resulting identities generalize earlier single-sum versions obtained by Andrews via Bailey pairs as well as subsequent refinements by Shi, Sun and Yao. A reader working in q-series would follow the argument because these double-sum forms enlarge the set of explicit product representations available for certain generating functions in partition theory.

Core claim

Substituting the expansion formula of Chebyshev polynomials in terms of q-Hermite polynomials into the orthogonality relations of the q-Hermite polynomials produces a family of double-sum identities of Rogers-Ramanujan type that extend the single-sum identities previously derived from the same polynomials.

What carries the argument

The expansion formula of Chebyshev polynomials in terms of q-Hermite polynomials, which is substituted into the orthogonality relations of the q-Hermite polynomials to extract the double-sum identities.

If this is right

  • The double-sum identities contain the single-sum Rogers-Ramanujan identities of Andrews as special cases obtained by fixing one summation index.
  • The same substitution technique supplies further identities whenever additional parameters are introduced into the Chebyshev or q-Hermite families.
  • The resulting q-series can be rewritten in product form, supplying new explicit representations for certain generating functions studied in partition theory.
  • The method extends the Bailey-pair constructions used by Andrews to a broader setting that includes double sums.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The double-sum structure may permit iterative applications of the same orthogonality step to produce triple-sum or higher identities.
  • Because q-Hermite polynomials appear in models of quantum algebras, the new identities could supply explicit summation formulas for matrix elements or partition functions in those models.

Load-bearing premise

The known expansion of Chebyshev polynomials in q-Hermite polynomials can be substituted directly into the orthogonality relations to produce the claimed double-sum identities without hidden restrictions on the ranges of summation or convergence.

What would settle it

Evaluating both sides of one of the derived double-sum identities at a concrete value of q (for example q = 1/2) and with the outer sums truncated at a modest upper limit, then checking whether the numerical values agree to within the truncation error.

read the original abstract

As the $q$-analog of Chebyshev polynomials, $q$-Hermite polynomials form a cornerstone in the family of $q$-orthogonal polynomials, which play a fundamental role in quantum algebra and mathematical physics. Recently, Andrews obtained a series of Rogers-Ramanujan type identities by constructing Bailey pairs from Chebyshev polynomials. In this paper, by applying the expansion formula of Chebyshev polynomials in terms of $q$-Hermite polynomials and using the orthogonality relations, we derive a series of Rogers-Ramanujan type identities on double sums, which further generalized the known results due to Andrews, Shi, Sun and Yao.

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

2 major / 0 minor

Summary. The paper claims that by substituting the known expansion formula expressing Chebyshev polynomials in terms of q-Hermite polynomials into the orthogonality relations for the q-Hermite polynomials, one obtains a family of Rogers-Ramanujan-type identities that involve double (rather than single) sums; these are presented as generalizations of earlier single-sum identities due to Andrews, Shi, Sun, and Yao.

Significance. If the derivations are complete and the summation interchanges are justified, the resulting double-sum identities would constitute a systematic extension of the Bailey-pair approach to RR identities, potentially supplying new generating-function tools in q-series and partition theory.

major comments (2)
  1. [Main derivation (following the statement of the expansion formula)] The central step that converts the single-index orthogonality relation into a double-sum identity requires an explicit reindexing of the double sum together with a justification for interchanging the order of summation (or for the absence of remainder terms). No such bounds on the summation indices or convergence argument appear in the derivation.
  2. [Statements of the new identities] The manuscript provides neither explicit summation limits for the resulting double sums nor any numerical verification or special-case checks against known single-sum RR identities, leaving the correctness of the claimed closed forms only partially supported.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised highlight areas where additional rigor and explicit details will strengthen the presentation. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: The central step that converts the single-index orthogonality relation into a double-sum identity requires an explicit reindexing of the double sum together with a justification for interchanging the order of summation (or for the absence of remainder terms). No such bounds on the summation indices or convergence argument appear in the derivation.

    Authors: We agree that the original derivation omitted an explicit reindexing and convergence justification. In the revised manuscript we have inserted the missing reindexing steps and added a dedicated paragraph establishing absolute convergence of the q-series for |q|<1. This justifies the interchange of summation order with no remainder terms, and the summation indices are now stated with explicit bounds in the derivation. revision: yes

  2. Referee: The manuscript provides neither explicit summation limits for the resulting double sums nor any numerical verification or special-case checks against known single-sum RR identities, leaving the correctness of the claimed closed forms only partially supported.

    Authors: We accept that explicit limits and verification examples improve the manuscript. The revised version now states the precise summation limits for every double-sum identity. We have also added a short subsection containing numerical checks for representative values of q and n, together with special-case reductions that recover the single-sum Rogers-Ramanujan identities of Andrews, Shi, Sun, and Yao. revision: yes

Circularity Check

0 steps flagged

Derivation relies on externally established q-Hermite properties with no reduction to inputs

full rationale

The paper states that the double-sum identities are obtained by substituting the known expansion formula of Chebyshev polynomials into the orthogonality relations of q-Hermite polynomials. These formulas are standard, pre-existing results in q-orthogonal polynomial theory and are not defined using the target identities. The work generalizes prior single-sum results by Andrews, Shi, Sun and Yao but does not use those citations to justify the core steps or forbid alternatives. No equation in the described chain reduces by construction to a fitted parameter, self-definition, or self-citation chain; the derivation remains independent of the claimed outputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivations rely on the standard expansion relation between Chebyshev and q-Hermite polynomials together with the orthogonality of the q-Hermite family; both are established background results in the literature on q-orthogonal polynomials.

axioms (2)
  • standard math Expansion formula expressing Chebyshev polynomials in the q-Hermite basis
    Invoked to convert the Chebyshev side into a sum over q-Hermite terms before orthogonality is applied.
  • standard math Orthogonality relations for q-Hermite polynomials
    Used to isolate coefficients and produce the double-sum identities.

pith-pipeline@v0.9.0 · 5391 in / 1327 out tokens · 73547 ms · 2026-05-08T08:14:22.084967+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages

  1. [1]

    Al-Salam and M.E.H

    W.A. Al-Salam and M.E.H. Ismail,q-beta integrals andq-Hermite poly- nomials, Pacific J. Math.135(2) (1988) 209–221

    work page 1988

  2. [2]

    Andrews, An analytic generalization of the Rogers–Ramanujan identities for odd moduli

    G.E. Andrews, An analytic generalization of the Rogers–Ramanujan identities for odd moduli. Proc. Natl. Acad. Sci. USA71(1974) 4082– 4085

    work page 1974

  3. [3]

    Andrews, Ramanujan’s “lost” notebook, I: partial theta functions, Adv

    G.E. Andrews, Ramanujan’s “lost” notebook, I: partial theta functions, Adv. Math.41(1981) 137–172

    work page 1981

  4. [4]

    Andrews, The Theory of Partitions, Cambridge University, Cam- bridge, 1984

    G.E. Andrews, The Theory of Partitions, Cambridge University, Cam- bridge, 1984

    work page 1984

  5. [5]

    Andrews,q-Orthogonal polynomials, Rogers–Ramanujan identi- ties, and mock theta functions, Proc

    G.E. Andrews,q-Orthogonal polynomials, Rogers–Ramanujan identi- ties, and mock theta functions, Proc. Steklov Inst. Math.276(1) (2012) 21–32. 22

    work page 2012

  6. [6]

    Favorite

    G.E. Andrews, Dyson’s “Favorite” identity and Chebyshev polynomials of the third and fourth kind, Ann. Comb.23(2019) 443–464

    work page 2019

  7. [7]

    Andrews and B.C.Berndt, Ramanujan’s Lost Notebook, Part II, Springer, New York, 2009

    G.E. Andrews and B.C.Berndt, Ramanujan’s Lost Notebook, Part II, Springer, New York, 2009

    work page 2009

  8. [8]

    Andrews and A.K

    G.E. Andrews and A.K. Uncu, Sequences in overpartitions, Ramanujan J.61(2) (2023) 715–727

    work page 2023

  9. [9]

    Askey and J

    R. Askey and J. Wilson, Some basic hypergeometric orthogonal polyno- mials that generalize Jacobi polynomials, Mem. Amer. Math. Soc.54 (1985), no.319, iv+55 pp

    work page 1985

  10. [10]

    Bailey, Some identities in combinatory analysis, Proc

    W.N. Bailey, Some identities in combinatory analysis, Proc. London Math. Soc.49(1947) 421–425

    work page 1947

  11. [11]

    Bailey, Identities of the Rogers–Ramanujan type, Proc

    W.N. Bailey, Identities of the Rogers–Ramanujan type, Proc. Lond. Math. Soc.50(2) (1949) 1–10

    work page 1949

  12. [12]

    Cao and L.Q

    Z.N. Cao and L.Q. Wang, Multi–sum Rogers–Ramanujan type identities, J. Math. Anal. Appl.522(2) (2023) pp. 24

    work page 2023

  13. [13]

    Chen and Z.G

    D.D. Chen and Z.G. Liu, New proofs of theorems onq-orthogonal func- tions, Int. J. Number Theory21(2) (2025) 473–486

    work page 2025

  14. [14]

    Chu and C.Y

    W.C. Chu and C.Y. Wang, Iteration process for multiple Rogers– Ramanujan identities, Ukrainian Math. J.64(1) (2012) 110–139

    work page 2012

  15. [15]

    Garrett, M.E.H

    K. Garrett, M.E.H. Ismail and D. Stanton, Variants of the Rogers– Ramanujan identities, Adv. Appl. Math.23(1999) 274–299

    work page 1999

  16. [16]

    Gasper and M

    G. Gasper and M. Rahman, Basic Hypergeometric Series, 2nd ed., Cam- bridge University, Cambridge, 2004

    work page 2004

  17. [17]

    Gordon, A combinatorial generalization of the Rogers–Ramanujan identities, Amer

    B. Gordon, A combinatorial generalization of the Rogers–Ramanujan identities, Amer. J. Math.83(1961) 393–399

    work page 1961

  18. [18]

    Ismail and D.R

    M.E.H. Ismail and D.R. Masson,q-Hermite polynomials, biorthogonal rational functions, andq-beta integrals, Trans. Amer. Math. Soc.346 (1994) 63–116. 23

    work page 1994

  19. [19]

    Kanade and M.C

    S. Kanade and M.C. Russell, Staircases to analytic sum–sides for many new integer partition identities of Rogers-Ramanujan type, Electron. J. Comb.26(2019) 1–6

    work page 2019

  20. [20]

    Laughlin, A.V

    J.M. Laughlin, A.V. Sills and P. Zimmer, Rogers–Ramanujan–Slater type identities. Electron. J. Combin.15(2008) pp. 59

    work page 2008

  21. [21]

    MacMahon, Combinatory Analysis

    P.A. MacMahon, Combinatory Analysis. vol. 2, Cambridge University, New York, 1916

    work page 1916

  22. [22]

    Mason and D.C

    J.C. Mason and D.C. Handscomb, Chebyshev Polynomials, Chapman and Hall, New York, 2003

    work page 2003

  23. [23]

    Nahm, Conformal field theory and the dilogarithm, in: 11th Interna- tional Conference on Mathematical Physics (Paris 1994), 662–667, Int

    W. Nahm, Conformal field theory and the dilogarithm, in: 11th Interna- tional Conference on Mathematical Physics (Paris 1994), 662–667, Int. Press, Cambridge, 1994

    work page 1994

  24. [24]

    Nahm, Conformal field theory, dilogarithms and three dimensional manifold, in: Interface Between Physics and Mathematics (Hangzhou 1993), 154–165, World Scientific, Singapore, 1994

    W. Nahm, Conformal field theory, dilogarithms and three dimensional manifold, in: Interface Between Physics and Mathematics (Hangzhou 1993), 154–165, World Scientific, Singapore, 1994

    work page 1993

  25. [25]

    Nahm, Conformal field theory and torsion elements of the Bloch group, in: Frontiers in Number Theory, Physics and Geometry, II, 67– 132, Springer, Berlin, 2007

    W. Nahm, Conformal field theory and torsion elements of the Bloch group, in: Frontiers in Number Theory, Physics and Geometry, II, 67– 132, Springer, Berlin, 2007

    work page 2007

  26. [26]

    Rogers, On a three–fold symmetry in the elements of Heine’s series, Proc

    L.J. Rogers, On a three–fold symmetry in the elements of Heine’s series, Proc. Lond. Math. Soc.24(1893) 171–179

    work page

  27. [27]

    Rogers, Second memoir on the expansion of certain infinite prod- ucts, Proc

    L.J. Rogers, Second memoir on the expansion of certain infinite prod- ucts, Proc. Lond. Math. Soc.25(1894) 318–343

    work page

  28. [28]

    Rogers, On two theorems of combinatory analysis and some allied identities, Proc

    L.J. Rogers, On two theorems of combinatory analysis and some allied identities, Proc. Lond. Math. Soc.12(1917) 315–336

    work page 1917

  29. [29]

    Rogers and S

    L.J. Rogers and S. Ramanujan, Proof of certain identities in combinatory analysis, Proc. Cambridge Philos. Soc.19(1919) 211–216

    work page 1919

  30. [30]

    Sang and D.Y.H

    D.D.M. Sang and D.Y.H. Shi, An Andrews–Gordon type identity for overpartitions, Ramanujan J.37(3) (2015) 653–679

    work page 2015

  31. [31]

    Sills, Finite Rogers–Ramanujan type identities, Electron

    A.V. Sills, Finite Rogers–Ramanujan type identities, Electron. J. Com- bin.10(2003) pp. 122. 24

    work page 2003

  32. [32]

    Slater, A new proof of Rogers’s transformations of infinite series, Proc

    L.J. Slater, A new proof of Rogers’s transformations of infinite series, Proc. Lond. Math. Soc.53(1951) 460–475

    work page 1951

  33. [33]

    Slater, Further identities of the Rogers–Ramanujan type, Proc

    L.J. Slater, Further identities of the Rogers–Ramanujan type, Proc. Lond. Math. Soc.54(1952) 147–167

    work page 1952

  34. [34]

    Sun, Rogers–Ramanujan type identities and Chebyshev polynomi- als of the third kind, Ramanujan J.60(3) (2023) 761–794

    L.H. Sun, Rogers–Ramanujan type identities and Chebyshev polynomi- als of the third kind, Ramanujan J.60(3) (2023) 761–794

    work page 2023

  35. [35]

    Schur, Ein Beitrag zur Additiven Zahlentheorie und zur Theorie der Kettenbr¨ uche, Sitzungsberichte der Berliner Akademie, (1917), 302–321

    I. Schur, Ein Beitrag zur Additiven Zahlentheorie und zur Theorie der Kettenbr¨ uche, Sitzungsberichte der Berliner Akademie, (1917), 302–321

    work page 1917

  36. [36]

    Szeg¨ o, Ein Beitrag zur Theorie der Thetafunctionen, Sitz

    G. Szeg¨ o, Ein Beitrag zur Theorie der Thetafunctionen, Sitz. Preuss. Akad. Wiss. Phys. Math.19(1926) 242–251

    work page 1926

  37. [37]

    Wang, New proofs of some double sum Rogers-Ramanujan type identities, Ramanujan J.62(1) (2023) 251–272

    L.Q. Wang, New proofs of some double sum Rogers-Ramanujan type identities, Ramanujan J.62(1) (2023) 251–272

    work page 2023

  38. [38]

    Yao, Rogers–Ramanujan type identities and Chebyshev polyno- mials, Proc

    O.X.M. Yao, Rogers–Ramanujan type identities and Chebyshev polyno- mials, Proc. Amer. Math. Soc.153(3) (2025) 1215–1229. 25

    work page 2025