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Normal ordering in the $(p,q)$-deformed generalized Weyl algebra. I: Algebraic Framework and Combinatorial Identities

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Pith's one-line read The paper claims a fully explicit normal-ordered form for every word in the (p,q)-deformed generalized Weyl algebra, with coefficients built from twin-basic binomial symbols and generalized Stirling numbers.

desk verdict The generic (p,q) normal-ordering formulas collapse on a false Lemma 2.10; the framework is promising but the main theorem is wrong as stated. read the letter →

arxiv 2606.22585 v2 pith:K3LMQIVR submitted 2026-06-21 math.CO

Toufik Mansour , Lahcen Oussi , Matthias Schork This is my paper · ORCID
classification math.CO MSC 05A1911B7305A3081R99
keywords (pq)-deformedWeylalgebranormalorderingYoungdiagramsStirlingnumberstwin-basicq-binomialcoefficientsgeneralizedpartitionsums
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper aims to establish that, in the (p,q)-deformed generalized Weyl algebra, every word can be brought to normal ordered form with coefficients written in closed form. The claimed formula packages all commutation crossings into powers of p and q, products of twin-basic numbers, and a twin-basic binomial factor, covering the q-deformed Weyl algebra, shift algebra, and Jordan plane as special cases. If the formula is correct, it replaces step-by-step commutation with a single finite sum and expresses the algebra's generalized Stirling numbers directly rather than through a recurrence. A sympathetic reader would care because normal ordering in deformed Weyl algebras sits at the intersection of quantum operator calculus and combinatorial statistics, and a uniform closed formula would unify many scattered special cases.

What carries the argument

The load-bearing object is the Young-diagram sum in Lemma 2.10. Normal ordering is carried out by repeatedly applying the commutation XY - qYX = hY^s Z_p, and each application produces a partition whose boxes record how many times the commutation has been used. Lemma 2.10 collapses the sum over all Young diagrams inside an (m-k) by k box, weighted by p^{|λ|} q^{(s-1)|λ|}, into the single twin-basic binomial (m choose k)_{p,q^{s-1}}. That conversion is what turns the recursive Young-diagram expansion into the closed product formulas of Corollary 2.12, Proposition 2.13, and Theorem 3.1. The supporting functions P_r, Q_{r|s}, and A_{r|s;p,q} respectively keep track of the p-power, q-power, and

What would settle it

Evaluate Eq. (15) for p=2, q=3, s=2, m=2, k=1. The left-hand side sums over the two partitions in I_{1,1}: the empty partition with |λ|=0 and the partition 1 with |λ|=1, giving 1 + 2·3 = 7. The right-hand side is (2 choose 1)_{2,3} = [2]_{2,3} = 2+3 = 5. Since 7 ≠ 5, the identity that powers the clean normal-ordering formulas is not generally valid.

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Extended reading notes

Core claim

The paper's central claim, stated on its own terms, is Theorem 3.1: for the algebra A_{s;h|p,q} generated by X, Y, Z_p with XY - qYX = hY^s Z_p, XZ_p = pZ_p X, and Z_pY = pYZ_p, the normal-ordered form of any word ω = X^{m_r}Y^{n_r}...X^{m_1}Y^{n_1} is ω = sum over k in [0,m]^r of h^{|k|} p^{P_r(m,k)} q^{Q_{r|s}(m,n,k)} binom(m,k)_{p,q^{s-1}} A_{r|s;p,q}(n,k) Y^{|n|+(s-1)|k|} Z_p^{|k|} X^{|m|-|k|}. Here binom(m,k)_{p,q^{s-1}} is a quotient of twin-basic factorials built from [n]_{p,q} = (p^n - q^n)/(p-q), and the auxiliary functions P_r, Q_{r|s}, A_{r|s;p,q} track the p-powers, q-powers, and (p,q)-factorial factors accumulated while moving X's past Y's and Z_p's. For the special word (YX)^r,

Load-bearing premise

The entire closed-form structure rests on the unproved identity in Eq. (15): that the sum of p^{|λ|} q^{(s-1)|λ|} over Young diagrams in an (m-k) by k box equals the twin-basic binomial (m choose k)_{p,q^{s-1}}; for p and q both different from 1 and s different from 1, direct small cases contradict this identity, and if it fails, Corollary 2.12, Proposition 2.13, and Theorem 3.1 collapse.

Editorial extensions

If this is right

  • If Theorem 3.1 holds, every word in the algebra is normal ordered by a finite sum of products of explicitly known (p,q)-symbols, eliminating the need for recursive commutation.
  • The generalized Stirling numbers S_{s;h}(r,k|p,q), previously given only by a recurrence, acquire a direct closed expression as a sum over the 2^{r-1} binary strings of length r-1.
  • Setting p=1 recovers the q-deformed generalized Weyl algebra results, and setting s=0,1,2 recovers the structure constants for the (p,q)-Weyl algebra, the shift algebra, and the Jordan plane given in Section 2.3.
  • For s=1 the general formula factorizes, giving ω = Y^{|n|} times a product over j of (q^{Σ_{l≤j} n_l} X + h[Σ_{l≤j} n_l]_{p,q} Z_p)^{m_j}, a direct (p,q)-deformation of the shift algebra binomial rule.
  • In the h → 0 limit the formula reduces to ω = q^{I(ω)} Y^{|n|} X^{|m|}, the standard normal-ordering rule for q-commuting variables.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Because Lemma 2.10 is the only step that collapses Young-diagram sums into a single binomial coefficient, a direct check of that identity is the fastest way to test the whole framework; evaluating for p=2, q=3, s=2, m=2, k=1 already gives a contradiction, so the clean closed forms should be treated as conditional until the summation is repaired.
  • If the identity is repaired by replacing the twin-basic binomial with a genuine two-parameter Gaussian binomial or with a sum over restricted partitions, the coefficients in Theorem 3.1 would become nested sums; the generalized Stirling-number interpretation may survive, but the single-product form would not.
  • A concrete testable extension: compute the normal-ordered coefficient for a small word, such as X^2 Y^2 X Y, with generic parameters p,q,s using direct algebra or computer algebra, and compare against Eq. (35); a mismatch would pinpoint exactly which of P_r, Q_{r|s}, A_{r|s;p,q} needs adjustment.
  • The paper's announced sequel on (p,q)-rook numbers would inherit whatever correction the coefficients require; a rook-number model matching the true Young-diagram sums rather than the collapsed binomial may be the more natural combinatorial target.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This paper studies normal ordering in the associative algebra A_{s;h|p,q} generated by X, Y, Z_p with relations XY - qYX = hY^sZ_p, XZ_p = pZ_pX, and Z_pY = pYZ_p. The authors first give a Young-diagram expansion for X^mY for a general polynomial f(Y), then specialize to f(Y)=Y^s. They claim a closed form involving the twin-basic binomial coefficient \binom{m}{k}_{p,q^{s-1}} and extend it to X^mY^n, to the cases s=0,1,2,3, and finally to arbitrary words ω = X^{m_r}Y^{n_r}...X^{m_1}Y^{n_1} (Theorem 3.1). The coefficients are connected to generalized Stirling numbers. The claimed results would unify known deformed Weyl, shift, and Jordan-plane normal ordering formulas.

Significance. The Young-diagram method is natural and the manuscript is clearly organized; for p=1 it recovers known q-deformed results, and the special cases s=0,1 are useful. However, the central identity Lemma 2.10 is false for generic p, and the main normal-ordering formulas therefore fail for the very (p,q)-deformed setting advertised. The correct replacement is standard — the ordinary Gaussian binomial with base p q^{s-1} — so the framework may be repairable, but as submitted the claimed theorem is not valid.

major comments (3)
  1. [Section 2.1, Lemma 2.10 (Eq. 15)] The asserted identity is false under the paper's own definitions. For m=2, k=1, I_{1,1} has two diagrams with |λ|=0,1, so the left side is 1 + p q^{s-1}, while \binom{2}{1}_{p,q^{s-1}} = [2]_{p,q^{s-1}} = p + q^{s-1}. The correct generating function for partitions in an (m-k)×k box is the ordinary Gaussian binomial \binom{m}{k}_{p q^{s-1}}. Since Corollary 2.12 (Eq. (17)), Proposition 2.13 (Eq. (18)), the special cases (22), (25), (30), (33), and Theorem 3.1 (Eq. (35)) all use Lemma 2.10, all these formulas fail for generic p (and generic s≠1).
  2. [Section 2.3.1, Eq. (22)] The failure is not merely formal. For s=0, generic h, m=2, n=1, Eq. (22) gives X^2Y = q^2YX^2 + h q(p+q^{-1})Z_pX + ...; the k=1 coefficient is h(1+pq). Directly from (20), X^2Y = q^2YX^2 + h(q+p)Z_pX. These agree only if (1-p)(1-q)=0. Thus the normal-ordering formula contradicts the defining relations for generic p,q.
  3. [Section 3.1, Theorem 3.1 (Eq. 35)] The arbitrary-word formula depends on the same false binomial through Eq. (18); the induction proof in Theorem 3.1 has no valid base case for generic p, because the r=1 and r=2 cases already use the false Lemma. Consequently Proposition 3.7's expression (50) for the generalized Stirling numbers is also invalid in the advertised (p,q)-generality. The framework may be repairable by replacing every \binom{m}{k}_{p,q^{s-1}} with \binom{m}{k}_{p q^{s-1}}, but the manuscript as submitted does not contain the correct statement.
minor comments (4)
  1. [Section 2.3.1, Eq. (22)] The quotient [n]_{p,q}! / [n-k]_{p,q}! is undefined for k>n; the product form ∏_{j=0}^{k-1}[n-j]_{p,q} is clearer and automatically vanishes for k>n when [0]_{p,q}=0.
  2. [Section 3.1, proof of Theorem 3.1] In the displayed computation before Theorem 3.1, 'n3n2`n1' should read 'n3+n2+n1'; as written it is a typographical error.
  3. [Definition 2.11] The condition Z_1=I deserves a short explanation: for generic p, Z_p is a generator, while Z_1 should denote the specialization Z_{1}=I. The notation is understandable but can be confusing.
  4. [Example 2.5] The displayed values of g_d(λ;Y) for d=2,3 are presented without derivation; a sentence explaining that they follow from Definition 2.4 would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the central theorem is derived from the defining relations; the unproved (and likely false) Lemma 2.10 is a correctness defect, not a circular one.

full rationale

The derivation chain in this paper is direct rather than circular. Theorem 3.1 is obtained by induction from the defining relations (16), using Lemma 2.1, Proposition 2.2, Lemma 2.6, and Proposition 2.13, all of which are proved in the text from the commutation relations. The normal-ordering coefficients are not fitted to any target and are not defined in terms of the final output. The Young-diagram technique is adapted from the authors' earlier work [19], but the p,q,Z_p generalization is re-derived here with displayed arguments, not merely imported. The generalized Stirling numbers S_{s;h}(n,k|p,q) are taken from [23] as a definition, but Proposition 3.7 derives a new expression for them from Theorem 3.1 rather than assuming it. Self-citations to [19], [20], and [23] are contextual: they supply base cases, notation, and definitions, but the load-bearing induction is carried out in the present paper. The paper also recovers known p=1 cases, giving independent consistency checks. The serious issue is Lemma 2.10 (Eq. (15)): it is stated without proof and appears to be false for generic p,q,s, since the Young-diagram sum is the ordinary Gaussian binomial with base p q^{s-1}, not the twin-basic binomial binom(m,k)_{p,q^{s-1}}. If so, Corollary 2.12, Proposition 2.13, and Theorem 3.1 fail as stated. But this is a correctness or validity problem, not circularity: the identities do not hold by construction; they are simply wrong. Hence the circularity score is 0.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

No parameters are fitted to data; s,h,p,q are algebraic parameters of the defined algebra. The derivation's real burden is Lemma 2.10, an asserted-but-false partition-sum identity that carries the monomial specialization. The remaining axioms are standard definitions (twin-basic numbers, Young-diagram combinatorics) plus the domain assumption that the deformed algebra is well-defined.

assumptions (4)
  • domain assumption The defining relations (16) determine a consistent associative unital algebra A_{s;h|p,q}: XY-qYX=hY^sZ_p, Z_pY=pYZ_p, XZ_p=pZ_pX, Z_1=I.
    Postulated as the object of study (Definition 2.11); the paper assumes such an algebra exists and has the expected normal-ordered basis.
  • ad hoc to paper Lemma 2.10 partition-sum identity: sum_{λ∈I_{m-k,k}} p^{|λ|}q^{(s-1)|λ|} = binom(m,k)_{p,q^{s-1}}.
    Asserted without proof in Eq. (15) to collapse Young-diagram sums into a (p,q)-binomial; under the paper's own twin-basic definition it is false for generic p, and the correct statement requires the ordinary Gaussian binomial in pq^{s-1}.
  • standard math Twin-basic calculus: [n]_{p,q} = (p^n-q^n)/(p-q), factorial and binomial definitions, and the identities such as [k+1]_{p,q} = q^k + p[k]_{p,q} used in Lemma 2.1.
    Elementary definitions and algebraic identities used throughout; independent of the central claim.
  • standard math Young-diagram bijections λ ↔ λ* and λ ↔ λ_* partition I_{m+1-j,j} into the two image sets in the induction of Theorem 2.7.
    Standard bijective combinatorics of partitions inside an ℓ×k box; used in Lemma 2.6 and Theorem 2.7.
invented entities (1)
  • Generator Z_p (defining the algebra A_{s;h|p,q})
    purpose: Mediates the (p,q)-deformation: Z_1 = I recovers p=1 algebras; with X,Y it generalizes the Fibonacci operator and Oussi's (p,q)-structures to arbitrary words.
    Z_p is a formal symbol introduced through commutation relations (5)-(6). It is a mathematical generator, not an empirically motivated entity; its only handle is the algebra it defines, whose consistency is assumed.

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Pith. "Pith review of Normal ordering in the $(p,q)$-deformed generalized Weyl algebra. I: Algebraic Framework and Combinatorial Identities." pith.science (2026). https://pith.science/paper/K3LMQIVR

@misc{pith2026260622585,
  author       = {Pith},
  title        = {Pith review of: Normal ordering in the $(p,q)$-deformed generalized Weyl algebra. I: Algebraic Framework and Combinatorial Identities},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K3LMQIVR}},
  note         = {Machine review of arXiv:2606.22585}
}
abstract

The $(p,q)$-deformed generalized Weyl algebra is generated by variables $X, Y$ and $Z_p$ which satisfy the commutation relations $XY-qYX=h Y^sZ_{p}, XZ_p=pZ_pX$, and $Z_pY=pYZ_p$, with $s\in \mathbb{N}_0$. We investigate the problem of normal ordering arbitrary words in these letters with the help of Young diagrams, and we treat certain special cases explicitly. In particular, the connection to generalized Stirling numbers is considered in detail.

Figures

Figures reproduced from arXiv: 2606.22585 by the authors.

Figure 1
Figure 1. The Young diagrams in I4. Definition 2.4. Let fpyq “ řs k“0 αky k be a polynomial. For a partition λ with d ě 1 parts, we define gdpλ; Y q ” gdpλ1, λ2, . . . , λd; Y q :“ ÿs i1,i2,...,id“0 ´ q řd j“1 ijλj ź d j“1 αij d ź´1 j“1 ri1 ` ¨ ¨ ¨ ` ij ` 1 ´ jsp,q¯ Y i1`i2`¨¨¨`id`1´d . Moreover, for d “ 0, we set g0pH; Y q :“ Y. Note that gdpλ; Y q depends on the polynomial f, but to simplify the notation, we do not indicate… view at source ↗

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Normal ordering in the $(p,q)$-deformed generalized Weyl algebra. III: The binomial formula

    math.CO 2026-07 accept novelty 5.5 of 10

    Normal-ordering coefficients of (X+Y)^n in the (p,q)-deformed generalized Weyl algebra equal sums of (p,q)-deformed s-rook numbers over Ferrers boards in rectangles.

  2. Normal ordering in the $(p,q)$-deformed generalized Weyl algebra. II: Interpretation in terms of rook placements

    math.CO 2026-07 unverdicted novelty 5.0 of 10

    Normal ordering in the (p,q)-deformed generalized Weyl algebra yields (p,q)-deformed s-rook numbers that give combinatorial interpretations of (p,q)-generalized Stirling numbers on staircase boards.

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Works this paper leans on

33 extracted references · cited by 2 Pith papers

  1. [2]

    Benkart, S.A

    G. Benkart, S.A. Lopes and M. Ondrus, A parametric family of subalgebras of the Weyl algebra. II: Irreducible modules, Recent developments in algebraic and combinatorial aspects of representation theory. Contemp. Math. 602, Amer. Math. Soc., Providence, RI (2013), 73–98

    2013

  2. [1]

    Beaudoin, G

    A. Beaudoin, G. Bergeron, A. Brillant, J Gaboriaud, L. Vinet and A. Zhedanov, Orthogonal polynomials and the deformed Jordan plane , J. Math. Anal. Appl. 507 (2022), Article 125717

    2022

  3. [3]

    Benkart, S.A

    G. Benkart, S.A. Lopes and M. Ondrus, A parametric family of subalgebras of the Weyl algebra. I: Structure and automorphisms, Trans. Am. Math. Soc. 367 (2015), 1993–2021

    2015

  4. [4]

    Benkart, S.A

    G. Benkart, S.A. Lopes and M. Ondrus, Derivations of a parametric family of subalgebras of the Weyl algebra , J. Algebra 424 (2015), 46–97

    2015

  5. [5]

    Blasiak and P

    P. Blasiak and P. Flajolet, Combinatorial models of creation-annihilation , Sém. Lothar. Combin. 65 (2011), Article B65c

    2011

  6. [6]

    Blumen, Two generalisations of the binomial theorem , Austral

    S.C. Blumen, Two generalisations of the binomial theorem , Austral. Math. Soc. Gaz. 33 (2006), 39–43

    2006

  7. [7]

    Burban and A.U

    I.M. Burban and A.U. Klimyk, pP, Qq-differentiation, pP, Qq-integration, and pP, Qq-hypergeometric functions re- lated to quantum groups , Integral Transforms Spec. Funct. 2 (1994), 15–36

    1994

  8. [8]

    Burde, On the matrix equation XA ´ AX “ X p, Linear Algebra Appl

    D. Burde, On the matrix equation XA ´ AX “ X p, Linear Algebra Appl. 404 (2005), 147–165

    2005

Show all 33 references
  1. [9]

    Celeste, R.B

    R.O. Celeste, R.B. Corcino and K.J.M. Gonzales, Two approaches to normal order coefficients , J. Integer Seq. 20 (2017), Art. 17.3.5

    2017

  2. [10]

    Gaddis, Two-parameter analogs of the Heisenberg enveloping algebra , Commun

    J. Gaddis, Two-parameter analogs of the Heisenberg enveloping algebra , Commun. Algebra 44 (2016), 4637–4653

    2016

  3. [11]

    Jagannathan, pP, Qq-special functions , in: Special functions and differential equations

    R. Jagannathan, pP, Qq-special functions , in: Special functions and differential equations. Proceedings of a work- shop, WSSF ’97, Madras, India, January 13–24, 1997, 158–164

    1997

  4. [12]

    Katriel and M

    J. Katriel and M. Kibler, Normal ordering for deformed boson operators and operator-valued deformed Stirling numbers, J. Phys. A: Math. Gen. 25 (1992), 2683–2691

    1992

  5. [13]

    Kirkman and L.W

    E.E. Kirkman and L.W. Small, q-analogs of harmonic oscillators and related rings , Israel J. Math. 81 (1993), 111–127

    1993

  6. [14]

    Koelink, Addition formulas for q-special functions

    H.T. Koelink, Addition formulas for q-special functions . In: Proceedings Special Functions, q-Series and Related Topics, Toronto, June (1995)

    1995

  7. [15]

    Koornwinder, Special functions and q-commuting variables , Fields Inst

    T.H. Koornwinder, Special functions and q-commuting variables , Fields Inst. Commun. 14 (1997), 131–166

    1997

  8. [16]

    Manin, Quantum Groups and Noncommutive Geometry , Centre de Recherches Mathématiques, Montréal (1988)

    Y.I. Manin, Quantum Groups and Noncommutive Geometry , Centre de Recherches Mathématiques, Montréal (1988)

    1988

  9. [17]

    Mansour, L

    T. Mansour, L. Oussi and M. Schork, Normal ordering in the pp, qq-deformed generalized Weyl algebra. II: Inter- pretation in terms of rook placements , Preprint (2026)

    2026

  10. [18]

    Mansour, L

    T. Mansour, L. Oussi and M. Schork, Normal ordering in the pp, qq-deformed generalized Weyl algebra. III: The binomial formula , Preprint (2026)

    2026

  11. [19]

    Mansour and M

    T. Mansour and M. Schork, The commutation relation xy “ qyx ` hf pyq and Newton ’s binomial formula , Ra- manujan J. 25 (2011), 405–445

    2011

  12. [20]

    Mansour, M

    T. Mansour, M. Schork, Commutation relations, normal ordering, and Stirling numbers , CRC Press, Boca Raton, FL (2016)

    2016

  13. [21]

    Mason and D.C

    J.C. Mason and D.C. Handscomb, Chebyshev Polynomials , Chapman and Hall/CRC, 2003

    2003

  14. [22]

    Oussi, A pp, qq-deformed recurrence for the Bell numbers , J

    L. Oussi, A pp, qq-deformed recurrence for the Bell numbers , J. Integer Seq. 23 (2020), Article 20.5.2

    2020

  15. [23]

    Oussi, pp, qq-analogues of the generalized Touchard polynomials and Stirling numbers , Indag

    L. Oussi, pp, qq-analogues of the generalized Touchard polynomials and Stirling numbers , Indag. Math. 33 (2022), 664–681

    2022

  16. [24]

    Oussi, A note on the pp, qq-derivative operator, Int

    L. Oussi, A note on the pp, qq-derivative operator, Int. J. Appl. Comput. Math. 10 (2024), Article 172

    2024

  17. [25]

    Potter, On the latent roots of quasi-commutative matrices , Amer

    H.S.A. Potter, On the latent roots of quasi-commutative matrices , Amer. Math. Monthly 57 (1950), 321–322

    1950

  18. [26]

    Sack, Taylor’s theorem for shift operators , Philos

    R.A. Sack, Taylor’s theorem for shift operators , Philos. Mag. VIII (1958), 497–503

    1958

  19. [27]

    Sau, Quantum conjugate momentum of angular momentum modulus , J

    J. Sau, Quantum conjugate momentum of angular momentum modulus , J. Phys. A. 11 (1978), 69–79

    1978

  20. [28]

    Schork, Recent developments in combinatorial aspects of normal ordering , Enumer

    M. Schork, Recent developments in combinatorial aspects of normal ordering , Enumer. Combin. Appl. 1 (2021), Article S2S2

    2021

  21. [29]

    Schützenberger, Une interprétation de certains solutions de l’équation fonctionnelle: F px ` yq “ F pxqF pyq, C

    M.P. Schützenberger, Une interprétation de certains solutions de l’équation fonctionnelle: F px ` yq “ F pxqF pyq, C. R. Acad. Sci. Paris 236 (1953), 352–353

    1953

  22. [30]

    Varvak, Rook numbers and the normal ordering problem , J

    A. Varvak, Rook numbers and the normal ordering problem , J. Combin. Theory Ser. A. 112 (2005), 292–307

    2005

  23. [31]

    Wachs and D

    M. Wachs and D. White, p, q-Stirling numbers and set partition statistics , J. Combin. Theory Ser. A. 56 (1991), 27–46

    1991

  24. [32]

    Wilcox, Exponential operators and parameter differentiation in quantum physics , J

    R.M. Wilcox, Exponential operators and parameter differentiation in quantum physics , J. Math. Phys. 8 (1967), 962–-982. 24 T. MANSOUR, L. OUSSI, AND M. SCHORK

    1967

  25. [33]

    Witschel, Ordered operator expansions by comparison , J

    W. Witschel, Ordered operator expansions by comparison , J. Phys. A. 8 (1974), 143–154. 1 Department of Mathematics, University of Haifa, 3498838 Haifa, Israel, tmansour@univ.haifa.ac.il 2 F aculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, ...

    1974

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