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arxiv: 2604.17498 · v1 · submitted 2026-04-19 · 🧮 math.FA

Exploring q-Stancu Operators via a New Representation

Feride Baraner , Ovgu Gurel This is my paper

Pith reviewed 2026-05-10 05:21 UTC · model grok-4.3

classification 🧮 math.FA
keywords q-Stancu operatorsq-Pochhammer symbolmomentsrecurrence relationsuniform convergencelimit operatorsq-Bernstein operators
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The pith

A representation of q-Stancu operators via the q-Pochhammer symbol yields a general recurrence for moments and proves uniform convergence of the limit form.

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

The paper develops a new expression for q-Stancu operators, which extend q-Bernstein operators, written in terms of the q-Pochhammer symbol. From this form the authors recover earlier properties and obtain a recurrence that expresses any higher-order moment in terms of lower-order moments. They introduce the limit operator obtained by letting the parameter approach its boundary value and prove that the original operators converge uniformly to it. The same recurrence is shown to hold for the moments of the limit operator. These results matter because moment relations simplify explicit calculations that would otherwise require direct summation over the operator definition for each order.

Core claim

The q-Stancu operators admit a representation in terms of the q-Pochhammer symbol. This representation immediately yields a general recurrence relation among the moments, so that the moment of order k+1 is determined by the moment of order k together with lower-order terms. The limit operator is defined by passing to the appropriate boundary value of the parameter; uniform convergence of the q-Stancu operators to this limit is established on the unit interval. The moments of the limit operator satisfy an analogous recurrence.

What carries the argument

The representation of the q-Stancu operators expressed directly in terms of the q-Pochhammer symbol, which converts the usual summation definition into a closed product form that simplifies moment calculations.

If this is right

  • Any higher-order moment of a q-Stancu operator can be obtained recursively from the moments of lower order.
  • The limit operator is reached by uniform convergence on the compact interval.
  • The moments of the limit operator obey the same recurrence pattern as those of the original operators.
  • Known identities for q-Stancu operators are recovered as immediate corollaries of the new representation.

Where Pith is reading between the lines

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

  • The recurrence may reduce the computational cost of evaluating high-order moments when these operators are used for function approximation.
  • Similar product representations could be sought for other q-analogues of positive linear operators to obtain parallel moment relations.
  • The uniform-convergence result supplies a concrete rate estimate once the difference between consecutive moments is bounded.

Load-bearing premise

The stated representation of the q-Stancu operators in terms of the q-Pochhammer symbol is valid and sufficient to derive the recurrence relations and the uniform-convergence result without additional restrictions on the parameters.

What would settle it

Direct evaluation of the third moment of the q-Stancu operator from its original summation definition and comparison with the value predicted by the recurrence applied to the first and second moments would confirm or refute the claimed relation.

read the original abstract

This paper investigates the q-Stancu operators, which generalize the q-Bernstein operators, by developing a new representation in terms of the q-Pochhammer symbol. Based on this representation, some known properties are re-discovered, and a general recurrence relation for the moments is established. It is shown that higher-order moments can be expressed in terms of lower-order ones. Furthermore, the limit form of the operators is defined and their uniform convergence is proved. Finally, the moments of the limit operator and their recurrence relations are presented.

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 / 3 minor

Summary. The paper develops a new representation of the q-Stancu operators in terms of the q-Pochhammer symbol. Using this representation, known properties are recovered, a general recurrence relation for the moments is derived (expressing higher-order moments via lower-order ones), the limit form of the operators is introduced, uniform convergence is proved, and the moments together with their recurrence relations for the limit operator are presented.

Significance. If the representation and derivations are valid, the work offers a streamlined approach to moment calculations and convergence analysis for q-Stancu operators, which generalize q-Bernstein operators. The recurrence relations constitute a practical and theoretical strength, enabling efficient computation of moments and potential extensions in q-approximation theory. The uniform convergence result for the limit operator aligns with standard techniques in the field and adds to the body of results on q-analogues of positive linear operators.

minor comments (3)
  1. The abstract states that 'some known properties are re-discovered' but does not list them; explicitly enumerating the recovered properties (e.g., in §2 or a dedicated subsection) would improve readability and allow readers to quickly assess the scope of the new representation.
  2. The domains and parameter restrictions (e.g., on q, n, and the Stancu parameter) under which the new q-Pochhammer representation holds and the recurrence/convergence results apply should be stated clearly at the outset, ideally in the introduction or before the main theorems.
  3. Notation for the q-Pochhammer symbol and the operators should be introduced with a brief reminder of standard definitions to ensure the paper is self-contained for readers outside the immediate q-calculus community.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the recognition of the utility of the new q-Pochhammer representation, the recurrence relations for moments, and the uniform convergence result for the limit operator. The recommendation for minor revision is noted. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper starts from a new representation of the q-Stancu operators expressed via the q-Pochhammer symbol. It then re-derives known properties, obtains a general recurrence relating higher-order moments to lower-order ones, defines the limit operator, and proves uniform convergence. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the recurrence and convergence proofs are presented as consequences of the new representation. The derivation chain is self-contained against external benchmarks in q-approximation theory, with no evidence of renaming known results as novel or importing uniqueness theorems from the authors' prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard properties of the q-Pochhammer symbol and q-analogues of classical operators; no free parameters, invented entities, or ad-hoc axioms are mentioned in the abstract.

axioms (1)
  • standard math Standard algebraic properties of the q-Pochhammer symbol hold and can be used to rewrite q-Stancu operators.
    Invoked to obtain the new representation described in the abstract.

pith-pipeline@v0.9.0 · 5375 in / 986 out tokens · 77600 ms · 2026-05-10T05:21:00.222356+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages

  1. [1]

    Agratini,On aq-analogue of Stancu operators, Cent

    O. Agratini,On aq-analogue of Stancu operators, Cent. Eur. J. Math.8(2010), 191–198

    work page 2010

  2. [2]

    G. E. Andrews, R. Askey, R. Roy,Special functions, Encyclopedia of Mathematics and Its Applications, Cambridge Univ. Press, Cambridge, 1999

    work page 1999

  3. [3]

    Eren¸ cin, G

    A. Eren¸ cin, G. Ba¸ scanbaz-Tunca, F. Ta¸ sdelen,Kantorovich typeq-Bernstein-Stancu oper- ators, Studia Univ. Babe¸ s-Bolyai Math.57(2012), no. 1, 89–105

    work page 2012

  4. [4]

    Finta,Approximation by Stancu typeq-operators, Ann

    Z. Finta,Approximation by Stancu typeq-operators, Ann. Univ. Ferrara62(2016), 217– 230

    work page 2016

  5. [5]

    Gupta, T

    V. Gupta, T. M. Rassias, H. Sharma,q-Durrmeyer operators based on P´ olya distribution, J. Nonlinear Sci. Appl.9(2016), no. 4, 1497–1504

    work page 2016

  6. [6]

    Il’inskii, S

    A. Il’inskii, S. Ostrovska,Convergence of generalized Bernstein polynomials, J. Approx. Theory116(2002), no. 1, 100–112. 19

    work page 2002

  7. [7]

    Jiang, J

    Y. Jiang, J. Li,The rate of convergence ofq-Bernstein-Stancu polynomials, Int. J. Wavelets Multiresolut. Inf. Process.7(2009), no. 6, 773–779

    work page 2009

  8. [8]

    V. Kac, P. Cheung,Quantum calculus, Springer, New York, 2001

    work page 2001

  9. [9]

    Lupa¸ s,Aq-analogue of the Bernstein operator, Seminar on Numerical and Statistical Calculus, Univ

    A. Lupa¸ s,Aq-analogue of the Bernstein operator, Seminar on Numerical and Statistical Calculus, Univ. Cluj-Napoca, No. 9 (1987)

    work page 1987

  10. [10]

    N. I. Mahmudov,Higher order limitq-Bernstein operators, Math. Methods Appl. Sci.34 (2011), no. 13, 1618–1626

    work page 2011

  11. [11]

    T. Neer, A. M. Acu, P. Agrawal,Approximation of functions by bivariateq-Stancu- Durrmeyer type operators, Math. Commun.23(2018), no. 2, 161–180

    work page 2018

  12. [12]

    Nowak,Approximation properties for generalizedq-Bernstein polynomials, J

    G. Nowak,Approximation properties for generalizedq-Bernstein polynomials, J. Math. Anal. Appl.350(2009), no. 1, 50–55

    work page 2009

  13. [13]

    Nowak, V

    G. Nowak, V. Gupta,The rate of pointwise approximation of positive linear operators based onq-integers, Ukr. Math. J.63(2011), 403–415

    work page 2011

  14. [14]

    Oru¸ c, G

    H. Oru¸ c, G. M. Phillips,A generalization of Bernstein polynomials, Proc. Edinburgh Math. Soc. (2)42(1999), no. 2, 403–413

    work page 1999

  15. [15]

    Ostrovska,q-Bernstein polynomials and their iterates, J

    S. Ostrovska,q-Bernstein polynomials and their iterates, J. Approx. Theory123(2003), no. 2, 232–255

    work page 2003

  16. [16]

    Ostrovska,On the Lupa¸ sq-analogue of the Bernstein operator, Rocky Mountain J

    S. Ostrovska,On the Lupa¸ sq-analogue of the Bernstein operator, Rocky Mountain J. Math. 36(2006), no. 5, 1615–1629

    work page 2006

  17. [17]

    Ostrovska, M

    S. Ostrovska, M. Turan,On the eigenvectors of theq-Bernstein operators, Math. Methods Appl. Sci.37(2014), no. 4, 562–570

    work page 2014

  18. [18]

    Ostrovska, M

    S. Ostrovska, M. Turan,On the block functions generating the limitq-Lupa¸ s operator, Quaest. Math.46(2023), no. 4, 711–719

    work page 2023

  19. [19]

    G. M. Phillips,Bernstein polynomials based on theq-integers, Ann. Numer. Math.4(1997), 511–518

    work page 1997

  20. [20]

    Turan,The truncated-Bernstein polynomials in the caseq >1, Abstr

    M. Turan,The truncated-Bernstein polynomials in the caseq >1, Abstr. Appl. Anal. 2014, Art. ID 126319

    work page 2014

  21. [21]

    V. S. Videnskii,On some classes ofq-parametric positive linear operators, Oper. Theory Adv. Appl.158(2005), 213–222. 20

    work page 2005

  22. [22]

    Y. Wang, Y. Zhou,Iterates properties forq-Bernstein-Stancu operators, Int. J. Model. Optim.3(2013), no. 4, 362–368

    work page 2013

  23. [23]

    Y. Wang, Y. Zhou,Shape preserving properties forq-Bernstein-Stancu operators, J. Math. 2014(2014), Art. ID 603694. 21

    work page 2014