Analytic summation of series involving higher-order derivatives of Chebyshev polynomials of the second kind and their applications to convolved linear recurrent sequences

Alexander Stokolos; Daniel Gray; Dmitriy Dmitrishin; Vitaly Khamitov

arxiv: 2605.03200 · v2 · submitted 2026-05-04 · 🧮 math.CV · math.CO· math.NT

Analytic summation of series involving higher-order derivatives of Chebyshev polynomials of the second kind and their applications to convolved linear recurrent sequences

Dmitriy Dmitrishin , Daniel Gray , Vitaly Khamitov , Alexander Stokolos This is my paper

Pith reviewed 2026-05-14 21:25 UTC · model grok-4.3

classification 🧮 math.CV math.COmath.NT

keywords Chebyshev polynomialsanalytic summationlinear recurrent sequencesFibonacci numbersLucas numbersPell numberscombinatorial identitiesclosed-form sums

0 comments

The pith

Series of higher-order derivatives of Chebyshev polynomials of the second kind sum analytically to rational functions expressed in the polynomials.

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

The paper shows that functional series whose terms are higher-order derivatives of Chebyshev polynomials of the second kind, with degree tied to derivative order, converge to explicit rational functions. These functions are written using the same Chebyshev polynomials evaluated at one fixed argument. A sympathetic reader would care because the closed forms immediately supply summation formulas for the series at concrete points and generate combinatorial identities for Fibonacci, Lucas, and Pell numbers together with their convolutions. The same expressions, continued analytically, also assign values to formally divergent series, recovering the classical Euler formulas in special cases.

Core claim

Analytic summation determines the rational functions to which these series converge; these functions are expressed in terms of Chebyshev polynomials evaluated at a specific argument, yielding new closed-form formulas for sums at various values and combinatorial identities for Fibonacci, Lucas, and Pell numbers and their convolutions. By analytic continuation the same functions supply sums for formally divergent series that recover Euler formulas in special cases.

What carries the argument

Analytic summation of series whose general term is the k-th derivative of the Chebyshev polynomial of the second kind of degree n, with n and k linked by a fixed linear relation, producing rational-function closed forms.

If this is right

The series admit explicit rational closed forms at every point inside the disk of convergence.
The closed forms generate summation identities for the Fibonacci, Lucas, and Pell sequences and for convolutions of these sequences.
Sections of the Fibonacci sequence satisfy new summation formulas derived from the same rational expressions.
Analytic continuation assigns finite values to formally divergent instances of the series, matching classical Euler formulas in limiting cases.

Where Pith is reading between the lines

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

The same summation technique could be applied to other classical orthogonal polynomial families whose generating functions are rational.
The resulting identities for recurrent-sequence convolutions may yield faster algorithms for computing partial sums of linear recurrences.
The explicit rational functions supply generating-function proofs for a wider class of combinatorial identities involving convolved sequences.

Load-bearing premise

The relation between polynomial degree and derivative order, together with the analytic properties of the Chebyshev polynomials, permits term-by-term differentiation and summation inside the disk of convergence without further justification for these series.

What would settle it

Numerical evaluation of partial sums of one such series at a concrete point strictly inside the disk of convergence, compared against the value of the claimed rational function at the same point.

read the original abstract

This paper considers functional series whose terms are higher-order derivatives of Chebyshev polynomials of the second kind, where the degree of the polynomial is related to the order of the derivative. Analytic summation is used to determine the rational functions to which these series converge. These functions are expressed in terms of Chebyshev polynomials evaluated at a specific argument. Connections are established between derivatives of Chebyshev polynomials of the second kind and special numerical sequences generated by linear recurrence relations. New closed-form formulas are obtained for the sums of the series at various values of the argument. As consequences, combinatorial identities are derived for the Fibonacci, Lucas, and Pell numbers, for sections of the Fibonacci sequence, and for their convolutions. By means of analytic continuation, sums of formally divergent series are obtained, which in special cases correspond to the classical Euler 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.

Desk Editor's Note private letter to a colleague

Closed forms for Chebyshev derivative series yield new identities for Fibonacci convolutions, but termwise summation needs better convergence proof.

read the letter

Hi, This paper finds closed forms for certain series of higher-order derivatives of Chebyshev polynomials of the second kind. The degree of the polynomial is chosen depending on the derivative order, and they sum these analytically to get rational functions expressed back in terms of Chebyshev polynomials. From there they pull out new summation formulas and combinatorial identities for Fibonacci numbers, Lucas numbers, Pell numbers, and convolutions of them. They also use analytic continuation to assign values to some formally divergent series, matching classical Euler sums in special cases. The work is solid on the combinatorial side. The identities are new as far as the abstract indicates, and linking the derivatives to the recurrent sequences through these sums is a clean move. It organizes some known material in a useful way for applications. Where it could be stronger is the core analytic step. The claim that the series converge to those rational functions relies on term-by-term operations inside the radius of convergence. Chebyshev polynomials grow exponentially with degree, and their derivatives grow with k!, so the linkage between degree and order needs to be checked carefully for uniform convergence or a dominating series. The stress-test note is right to flag this; if the paper doesn't supply an explicit majorant or invoke a theorem like Morera's or Weierstrass M-test for the specific series, that part of the argument has a gap. It's not fatal, but it should be addressed. The paper is aimed at researchers in special functions, orthogonal polynomials, and discrete mathematics who deal with generating functions and identities for linear recurrences. Someone looking for explicit formulas to compute or prove things about convolved sequences would get something out of it. I would bring this to a reading group to check the identities and see the derivations. It is worth sending to peer review, with the expectation that the authors add a paragraph or two on the convergence justification. Best,

Referee Report

2 major / 2 minor

Summary. The paper develops analytic summation techniques for functional series whose general term involves the k-th derivative of the Chebyshev polynomial of the second kind U_{n(k)}(x), with the degree n tied to the derivative order k. It obtains closed-form expressions for these sums as rational functions expressed in terms of Chebyshev polynomials evaluated at a fixed argument, derives new summation formulas at specific points, and extracts combinatorial identities for Fibonacci, Lucas, and Pell numbers together with their convolutions and sections. Analytic continuation is then applied to assign values to formally divergent series, recovering classical Euler-type formulas in special cases.

Significance. If the term-by-term operations and analytic continuation are rigorously justified, the results would supply explicit closed forms for a family of series linked to linear recurrences and yield new identities for well-studied integer sequences. The approach connects orthogonal-polynomial derivatives to recurrent-sequence convolutions in a potentially useful way, but the absence of explicit convergence controls limits immediate applicability.

major comments (2)

[§2–3 (analytic summation)] The central derivation (abstract and §2–3) performs term-by-term differentiation and summation of the series without an explicit majorant, appeal to the Weierstrass M-test on compact subsets of the disk, or Morera’s theorem. Given that |U_n(x)| grows like (2|x|)^n and the k-th derivative introduces a factorial factor in k while n = n(k), the justification for interchanging limits and derivatives inside the disk of convergence is missing and is load-bearing for all subsequent closed forms.
[final section (analytic continuation)] The analytic-continuation step that extends the closed forms to divergent series (final section) inherits the same gap: no separate verification is supplied that the rational-function expressions obtained inside the disk remain valid on the boundary or beyond via continuation.

minor comments (2)

[§1] Notation for the degree-order linkage n(k) is introduced without a displayed formula; a single displayed equation would clarify the relation for readers.
[§4] Several summation identities for Fibonacci and Pell convolutions are stated without cross-reference to the corresponding Chebyshev closed form; adding equation numbers would improve traceability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We agree that the manuscript requires additional explicit justification for the term-by-term operations and analytic continuation, and we will incorporate these in the revised version.

read point-by-point responses

Referee: [§2–3 (analytic summation)] The central derivation (abstract and §2–3) performs term-by-term differentiation and summation of the series without an explicit majorant, appeal to the Weierstrass M-test on compact subsets of the disk, or Morera’s theorem. Given that |U_n(x)| grows like (2|x|)^n and the k-th derivative introduces a factorial factor in k while n = n(k), the justification for interchanging limits and derivatives inside the disk of convergence is missing and is load-bearing for all subsequent closed forms.

Authors: We agree that the justification was insufficiently detailed. In the revised manuscript we will add a dedicated paragraph (or short subsection) in §2 that first recalls the explicit representation of U_n and its derivatives, derives a uniform bound |U_n^{(k)}(x)| ≤ C_k (2r)^n n^{O(k)} on any compact disk |x|≤r<1, and then invokes the Weierstrass M-test to justify uniform convergence of the differentiated series. Interchange of summation and differentiation will be further justified by appealing to the standard theorem on term-by-term differentiation of power series inside the disk of convergence (or, equivalently, by verifying the resulting function is holomorphic via Morera’s theorem on triangular contours). These additions will make the derivations rigorous while leaving the closed-form expressions unchanged. revision: yes
Referee: [final section (analytic continuation)] The analytic-continuation step that extends the closed forms to divergent series (final section) inherits the same gap: no separate verification is supplied that the rational-function expressions obtained inside the disk remain valid on the boundary or beyond via continuation.

Authors: We acknowledge the need for an explicit statement. In the revision we will add a short paragraph noting that each closed-form expression is a rational function of x (hence meromorphic on the whole plane). Because the rational function coincides with the sum of the series inside the disk of convergence, the identity theorem guarantees that it supplies the unique analytic continuation to the complement of its poles. On the boundary we will remark that the expressions remain valid by Abel summation whenever the series converges, recovering the classical Euler-type identities as special cases. This clarification will be inserted without altering the formal results. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation uses independent standard properties of Chebyshev polynomials

full rationale

The paper derives closed forms for the indicated series by analytic summation, expressing results as rational functions in Chebyshev polynomials of the second kind evaluated at a fixed argument. This rests on the known recurrence relations, generating functions, and analytic continuation properties of the U_n polynomials, none of which are defined in terms of the target sums. The links to Fibonacci, Lucas, Pell numbers and their convolutions are obtained by specializing the closed forms at particular arguments, using only the classical linear-recurrence definitions of those sequences. No equation reduces a derived sum to a fitted parameter or to a self-citation that itself depends on the present result; the derivation chain remains self-contained against external Chebyshev theory and does not invoke any of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The derivations rely on standard analytic properties of Chebyshev polynomials of the second kind and the theory of linear recurrence relations; no free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (2)

standard math Chebyshev polynomials of the second kind satisfy known recurrence and orthogonality relations that permit analytic continuation and summation of the indicated series.
Invoked implicitly when the paper states that the series converge to rational functions expressed via the same polynomials.
domain assumption Linear recurrent sequences such as Fibonacci numbers admit generating functions that can be manipulated via the same polynomial identities.
Used to translate the summation results into combinatorial identities.

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Analytic summation ... series ... higher-order derivatives of Chebyshev polynomials of the second kind ... converge to rational functions expressed in terms of Chebyshev polynomials evaluated at a specific argument
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat_equivNat unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Connections ... to Fibonacci, Lucas, and Pell numbers ... convolved ... k-sections

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

Abd-Elhameed and Ohood M

Waleed M. Abd-Elhameed and Ohood M. Alqubori,New results of unified chebyshev polynomials, AIMS Mathematics9(2024), no. 8, 20058–20088

work page 2024
[2]

8, 1175–1188

Leonid Bedratyuk and Nadiia Lunio,Derivations and identities for chebyshev polynomials, Ukrainian Mathematical Journal73(2022), no. 8, 1175–1188

work page 2022
[3]

Boyd,Chebyshev and fourier spectral methods, 2nd ed., Dover Publications, 2001, 688 p

John P. Boyd,Chebyshev and fourier spectral methods, 2nd ed., Dover Publications, 2001, 688 p

work page 2001
[4]

Cheves,Problem b-275, The Fibonacci Quarterly13(1975), no

W. Cheves,Problem b-275, The Fibonacci Quarterly13(1975), no. 1, 95

work page 1975
[5]

Dmytryshyn, A

D. Dmytryshyn, A. Stokolos, and D. Gray,Reciprocal polynomials with zeros on the unit circle and derivatives of chebyshev polynomials of the second kind, arXiv preprint arXiv:2602.19132, 2026

work page arXiv 2026
[6]

Hoggatt, V

Jr. Hoggatt, V. E. and D. A. Lind,Compositions and fibonacci num- bers, The Fibonacci Quarterly7(1969), no. 3, 253–266

work page 1969
[7]

Khamitov, D

V. Khamitov, D. Dmytryshyn, D. Gray, and A. Stokolos,Convolved numbers of k-section of the fibonacci sequence: properties, conse- quences, arXiv preprint arXiv:2603.08636v1, 2026

work page arXiv 2026
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Thomas Koshy,Fibonacci and lucas numbers with applications, Wiley-Interscience, 2001, 652 p

work page 2001
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Marcin Lawnik and Adrian Kapczy´ nski,Application of modified chebyshev polynomials in asymmetric cryptography, Computer Sci- ence20(2019), no. 3, 367–381

work page 2019
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1, 1156–1160

Helmut Prodinger,Representing derivatives of chebyshev polynomials by chebyshev polynomials and related questions, Open Mathematics 15(2017), no. 1, 1156–1160

work page 2017
[11]

Rivlin,Chebyshev polynomials: From approximation the- ory to algebra and number theory, 2nd ed., Wiley-Interscience / Dover Publications, 1990, 249 p

Theodore J. Rivlin,Chebyshev polynomials: From approximation the- ory to algebra and number theory, 2nd ed., Wiley-Interscience / Dover Publications, 1990, 249 p

work page 1990
[12]

Siyi Wang,Some new identities of chebyshev polynomials and their applications, Advances in Difference Equations2015(2015), no. 1, 355. 15

work page 2015

[1] [1]

Abd-Elhameed and Ohood M

Waleed M. Abd-Elhameed and Ohood M. Alqubori,New results of unified chebyshev polynomials, AIMS Mathematics9(2024), no. 8, 20058–20088

work page 2024

[2] [2]

8, 1175–1188

Leonid Bedratyuk and Nadiia Lunio,Derivations and identities for chebyshev polynomials, Ukrainian Mathematical Journal73(2022), no. 8, 1175–1188

work page 2022

[3] [3]

Boyd,Chebyshev and fourier spectral methods, 2nd ed., Dover Publications, 2001, 688 p

John P. Boyd,Chebyshev and fourier spectral methods, 2nd ed., Dover Publications, 2001, 688 p

work page 2001

[4] [4]

Cheves,Problem b-275, The Fibonacci Quarterly13(1975), no

W. Cheves,Problem b-275, The Fibonacci Quarterly13(1975), no. 1, 95

work page 1975

[5] [5]

Dmytryshyn, A

D. Dmytryshyn, A. Stokolos, and D. Gray,Reciprocal polynomials with zeros on the unit circle and derivatives of chebyshev polynomials of the second kind, arXiv preprint arXiv:2602.19132, 2026

work page arXiv 2026

[6] [6]

Hoggatt, V

Jr. Hoggatt, V. E. and D. A. Lind,Compositions and fibonacci num- bers, The Fibonacci Quarterly7(1969), no. 3, 253–266

work page 1969

[7] [7]

Khamitov, D

V. Khamitov, D. Dmytryshyn, D. Gray, and A. Stokolos,Convolved numbers of k-section of the fibonacci sequence: properties, conse- quences, arXiv preprint arXiv:2603.08636v1, 2026

work page arXiv 2026

[8] [8]

Thomas Koshy,Fibonacci and lucas numbers with applications, Wiley-Interscience, 2001, 652 p

work page 2001

[9] [9]

3, 367–381

Marcin Lawnik and Adrian Kapczy´ nski,Application of modified chebyshev polynomials in asymmetric cryptography, Computer Sci- ence20(2019), no. 3, 367–381

work page 2019

[10] [10]

1, 1156–1160

Helmut Prodinger,Representing derivatives of chebyshev polynomials by chebyshev polynomials and related questions, Open Mathematics 15(2017), no. 1, 1156–1160

work page 2017

[11] [11]

Rivlin,Chebyshev polynomials: From approximation the- ory to algebra and number theory, 2nd ed., Wiley-Interscience / Dover Publications, 1990, 249 p

Theodore J. Rivlin,Chebyshev polynomials: From approximation the- ory to algebra and number theory, 2nd ed., Wiley-Interscience / Dover Publications, 1990, 249 p

work page 1990

[12] [12]

Siyi Wang,Some new identities of chebyshev polynomials and their applications, Advances in Difference Equations2015(2015), no. 1, 355. 15

work page 2015