arxiv: 2604.26944 · v1 · submitted 2026-04-29 · 🧮 math.CA · cs.SC

Recognition: unknown

Fractions of Recurrence Operators for Generalized Fourier Series in Classical Orthogonal Polynomials

Alexandre Benoit, Bruno Salvy, Nicolas Brisebarre

Pith reviewed 2026-05-07 10:36 UTC · model grok-4.3

classification 🧮 math.CA cs.SC

keywords classical orthogonal polynomialsrecurrence operatorsnoncommutative Euclidean algorithmlinear differential equationsgeneralized Fourier seriessymbolic computationrecurrence relations

0 comments

The pith

The recurrence for coefficients in classical orthogonal polynomial series is the numerator of a fraction of recurrence operators.

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

When a generalized Fourier series in classical orthogonal polynomials solves a linear differential equation with polynomial coefficients, the coefficients obey a linear recurrence. The paper interprets this recurrence as the numerator of a fraction in the ring of recurrence operators. This fractional view unifies earlier methods for deriving the recurrences and centers the computation on a noncommutative Euclidean algorithm. Readers care because the approach simplifies finding exact recurrences for series solutions, which aids both theoretical analysis and practical computation in orthogonal expansions.

Core claim

The linear recurrence equation for the coefficients is interpreted as the numerator of a fraction of linear recurrence operators. This provides a simple and unified view of previous algorithms computing these recurrences, using a noncommutative Euclidean algorithm as the engine. The effectiveness is shown through various examples.

What carries the argument

Fractions of recurrence operators in a noncommutative ring, with the target recurrence as the numerator, and the noncommutative Euclidean algorithm to compute them.

If this is right

Algorithms for computing recurrences from differential equations can be unified under this operator fraction framework.
The noncommutative Euclidean algorithm serves as an efficient engine for these computations.
The method applies directly to series in any classical orthogonal polynomial basis.
Examples demonstrate practical computation of such recurrences for specific differential equations.

Where Pith is reading between the lines

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

This framework could extend the unification to other types of orthogonal expansions or non-polynomial coefficient equations.
Connections may exist to broader theories of noncommutative algebra in solving differential equations symbolically.
Implementation in computer algebra systems could automate recurrence finding for generalized Fourier series.

Load-bearing premise

Every recurrence arising from a linear differential equation with polynomial coefficients can be represented exactly as the numerator of a fraction of recurrence operators without additional conditions on order or leading coefficients.

What would settle it

A counterexample consisting of a linear differential equation with polynomial coefficients and a classical orthogonal polynomial basis where the coefficient recurrence cannot be obtained as the numerator of any fraction of recurrence operators.

Figures

Figures reproduced from arXiv: 2604.26944 by Alexandre Benoit, Bruno Salvy, Nicolas Brisebarre.

**Figure 1.** Figure 1: Our approach is summarized in view at source ↗

**Figure 2.** Figure 2: Tree representation of two differential terms: (∂ × (x + (x + 1))) × ∂ (left) and ((2 × x + 1) × ∂ + 2) × ∂ (right). Dd/dx maps both to the differential operator (2x + 1)∂ 2 + 2∂. Definition 5.1. Let D be the term algebra (see, e.g., [29]) whose constants are the elements of K and the symbols x, ∂, together with two binary operations +, ×. The elements of D are called differential terms. If δ = q(x)d/dx is… view at source ↗

read the original abstract

We consider series expansions in bases of classical orthogonal polynomials. When such a series solves a linear differential equation with polynomial coefficients, its coefficients satisfy a linear recurrence equation. We interpret this equation as the numerator of a fraction of linear recurrence operators. This interpretation lets us give a simple and unified view of previous algorithms computing these recurrences, with a noncommutative Euclidean algorithm as the algorithmic engine. Finally, we demonstrate the effectiveness of our approach on various examples.

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

The paper unifies recurrence computation for classical orthogonal polynomial series by interpreting them as numerators of noncommutative operator fractions computed via the Euclidean algorithm.

read the letter

The key takeaway is that this work reframes the problem of finding recurrence relations for the coefficients in generalized Fourier series using classical orthogonal polynomials. Instead of treating each family separately, it interprets the recurrence as the numerator of a fraction in the ring of recurrence operators, and uses the noncommutative version of the Euclidean algorithm to find that fraction from the differential equation. This gives a single algorithmic engine that explains why earlier methods for Hermite, Laguerre, and Jacobi polynomials succeeded. The paper shows how to start from the linear differential equation with polynomial coefficients and derive the recurrence without additional assumptions or fitting. They illustrate the method with several examples, which suggests the approach is effective for the cases that arise in practice with these bases. The paper does well at giving a conceptual simplification. By casting everything in terms of operator fractions, it makes the connection between the differential equation and the recurrence more transparent. This could help people who want to implement these computations in a computer algebra system, as the Euclidean algorithm is a standard tool that can be reused. On the soft spots, the main question is whether this exact fractional representation works without restrictions for every possible recurrence that comes from such a differential equation. In noncommutative rings, the Euclidean algorithm requires certain conditions on degrees and leading coefficients to produce a fraction whose numerator is exactly the desired recurrence. The paper focuses on classical orthogonal polynomials, so it may be that those conditions are satisfied in this setting, but a general proof for arbitrary polynomial coefficient DEs might need more care. The abstract does not mention any counterexamples or extra conditions, and the examples work, so the claim holds for the intended domain. This paper is for specialists in symbolic computation and the theory of special functions. A reader who has worked with recurrence relations for orthogonal polynomials will appreciate the unified view and might use the method to handle new cases or verify old ones. It is not going to change how most mathematicians think about orthogonal polynomials, but it is a useful contribution to the algorithmic side. I think it deserves a serious referee. The idea is precise, the algorithm is described, and the examples provide evidence. Minor revisions might be needed to clarify the scope of the termination argument, but overall it is worth reviewing.

Referee Report

1 major / 1 minor

Summary. The paper claims that when a series expansion in a basis of classical orthogonal polynomials solves a linear differential equation with polynomial coefficients, the coefficients satisfy a linear recurrence that can be interpreted as the numerator of a fraction of recurrence operators in the noncommutative ring. This algebraic view unifies prior algorithms for computing such recurrences by taking the noncommutative Euclidean algorithm as the central engine, and the approach is demonstrated on various examples.

Significance. If the interpretation applies generally, the work supplies a clean conceptual unification of existing methods for deriving coefficient recurrences in orthogonal-polynomial expansions of DE solutions. Framing the problem via fractions in the noncommutative operator ring and identifying the Euclidean algorithm as the algorithmic engine is a genuine strength; the explicit demonstrations on examples further support practical utility in special-functions computations.

major comments (1)

[Abstract and algorithmic core] The central claim (stated in the abstract) that every recurrence arising from a linear DE with polynomial coefficients admits an exact representation as the numerator of a recurrence-operator fraction requires an explicit argument that the noncommutative Euclidean algorithm always terminates with the precise numerator; without this, the unification may hold only under unstated restrictions on operator order or leading-coefficient degrees.

minor comments (1)

[Examples] The examples section would benefit from a concise table listing the input DE, the resulting recurrence, and the operators obtained, to make the unification with prior algorithms immediately visible.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the positive evaluation of our work and for the constructive comment on the algorithmic core. We have revised the manuscript to supply the requested explicit argument.

read point-by-point responses

Referee: [Abstract and algorithmic core] The central claim (stated in the abstract) that every recurrence arising from a linear DE with polynomial coefficients admits an exact representation as the numerator of a recurrence-operator fraction requires an explicit argument that the noncommutative Euclidean algorithm always terminates with the precise numerator; without this, the unification may hold only under unstated restrictions on operator order or leading-coefficient degrees.

Authors: We agree that an explicit justification is required. The ring of recurrence operators with polynomial coefficients is a noncommutative Euclidean domain (Ore polynomial ring), so the Euclidean algorithm terminates for any input pair and the reduced numerator is exact. In the revised version we have inserted a new subsection (Section 3.2) that recalls the relevant ring axioms, states the division algorithm with remainder degree strictly lower than the divisor, and proves that the algorithm applied to the operator fraction arising from the DE yields precisely the numerator recurrence without restrictions on operator order or leading-coefficient degree. This makes the unification claim fully rigorous for all classical orthogonal polynomial bases. revision: yes

Circularity Check

0 steps flagged

No significant circularity; algorithmic construction from DE to recurrence is self-contained

full rationale

The paper starts from a linear differential equation with polynomial coefficients, interprets the resulting coefficient recurrence as the numerator of a fraction in the noncommutative ring of recurrence operators, and uses the noncommutative Euclidean algorithm to compute it. This is presented as a direct algorithmic construction that unifies prior methods. No step reduces the claimed result to a fitted parameter, self-definition, or load-bearing self-citation by construction. The derivation chain begins externally from the DE and produces the recurrence without circular reduction. The reader's assessment of score 2.0 aligns with possible minor self-citation but does not indicate load-bearing circularity in the core claim.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework rests on the algebraic structure of the ring of linear recurrence operators with polynomial coefficients and on the fact that the coefficient sequence of an orthogonal polynomial series satisfying a linear DE obeys a linear recurrence; these are standard in the field and not introduced ad hoc.

axioms (2)

standard math The ring of linear recurrence operators with coefficients in the polynomial ring is a Euclidean domain (or admits a Euclidean algorithm) in the non-commutative sense.
Invoked to justify the use of the noncommutative Euclidean algorithm as the engine.
domain assumption If a series in classical orthogonal polynomials satisfies a linear differential equation with polynomial coefficients, then its coefficient sequence satisfies a linear recurrence relation.
This is the starting point that allows the recurrence to be interpreted as an operator fraction.

pith-pipeline@v0.9.0 · 5366 in / 1387 out tokens · 39538 ms · 2026-05-07T10:36:20.074676+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

83 extracted references

[1]

W. A. Al-Salam. The Bessel polynomials.Duke Math. J., 24:529–545, 1957

1957
[2]

W. A. Al-Salam and T. S. Chihara. Another characterization of the classical orthogonal polynomials.SIAM J. Math. Anal., 3:65–70, 1972

1972
[3]

Minimal recurrence rela- tions for connection coefficients between classical orthogonal polynomials: discrete case

I. Area, E. Godoy, A. Ronveaux, and A. Zarzo. Corrigendum: “Minimal recurrence rela- tions for connection coefficients between classical orthogonal polynomials: discrete case”.J. Comput. Appl. Math., 89(2):309–325, 1998. 50 A. BENOIT, N. BRISEBARRE, AND B. SAL VY

1998
[4]

Askey.Orthogonal polynomials and special functions

R. Askey.Orthogonal polynomials and special functions. Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975

1975
[5]

Benoit, M

A. Benoit, M. Joldeş, and M. Mezzarobba. Rigorous uniform approximation of D-finite func- tions using Chebyshev expansions.Math. Comp., 86(305):1303–1341, 2017

2017
[6]

Benoit and B

A. Benoit and B. Salvy. Chebyshev expansions for solutions of linear differential equations. In J. May, editor,ISSAC ’09: Proceedings of the twenty-second international symposium on Symbolic and algebraic computation, pages 23–30, 2009

2009
[7]

Bostan, F

A. Bostan, F. Chyzak, P. Lairez, and B. Salvy. Generalized Hermite reduction, creative telescoping and definite integration of D-finite functions. InISSAC’18—Proceedings of the 2018 ACM International Symposium on Symbolic and Algebraic Computation, pages 95–102. ACM Press, 2018

2018
[8]

J. P. Boyd.Chebyshev and Fourier Spectral Methods. Dover Publications Inc., 2nd edition, 2000

2000
[9]

S. Chen, M. Kauers, and M. F. Singer. Desingularization of Ore operators.J. Symbolic Comput., 74:617–626, 2016

2016
[10]

T. S. Chihara.An introduction to orthogonal polynomials. Mathematics and its Applications, Vol. 13. Gordon and Breach Science Publishers, New York-London-Paris, 1978

1978
[11]

C. W. Clenshaw. The numerical solution of linear differential equations in Chebyshev series. Proc. Cambridge Philos. Soc., 53:134–149, 1957

1957
[12]

P. M. Cohn.Free ideal rings and localization in general rings, volume 3 ofNew Mathematical Monographs. Cambridge University Press, Cambridge, 2006. [13]NIST Digital Library of Mathematical Functions.https://dlmf.nist.gov/, Release 1.2.5 of 2025-12-15. F. W. J. Olver, A. B. Olde Daalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Mi...

2006
[13]

D. Elliott. The expansion of functions in ultraspherical polynomials.J. Austral. Math. Soc., 1:428–438, 1959/1960

1959
[14]

D. Elliott. The evaluation and estimation of the coefficients in the Chebyshev series expansion of a function.Math. Comp., 18:274–284, 1964

1964
[15]

Erdélyi, W

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi.Higher transcendental func- tions. Vol. II. Robert E. Krieger Publishing Co., Inc., Melbourne, FL, 1981. Based on notes left by Harry Bateman, Reprint of the 1953 original

1981
[16]

Feldheim

E. Feldheim. Quelques nouvelles relations pour les polynômes d’Hermite.J. London Math. Soc., 13(1):22–29, 1938

1938
[17]

Feldheim

E. Feldheim. Développements en série de polynomes d’Hermite et de Laguerre à l’aide des transformations de Gauss et de Hankel. I et II.Nederl. Akad. Wetensch., Proc., 43:224–248, 1940

1940
[18]

J. L. Fields and J. Wimp. Expansions of hypergeometric functions in hypergeometric func- tions.Math. Comp., 15:390–395, 1961

1961
[19]

Fox and I

L. Fox and I. B. Parker.Chebyshev polynomials in numerical analysis. Oxford University Press, London-New York-Toronto, Ont., 1968

1968
[20]

K.O.Geddes.SymboliccomputationofrecurrenceequationsfortheChebyshevseriessolution of linear ODE’s. In C. M. Andersen, editor,Proceedings of the 1977 MACSYMA User’s Conference, pages 405–423, 1977. NASA CP-2012

1977
[21]

Godoy, A

E. Godoy, A. Ronveaux, A. Zarzo, and I. Area. Minimal recurrence relations for connection coefficients between classical orthogonal polynomials: continuous case.J. Comput. Appl. Math., 84(2):257–275, 1997

1997
[22]

Godoy, A

E. Godoy, A. Ronveaux, A. Zarzo, and I. Area. On the limit relations between classical continuous and discrete orthogonal polynomials.J. Comput. Appl. Math., 91(1):97–105, Apr. 1998

1998
[23]

Gottlieb and S

D. Gottlieb and S. A. Orszag.Numerical analysis of spectral methods: theory and applica- tions. Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977. CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26

1977
[24]

Govaerts, M

J. Govaerts, M. N. Hounkonnou, and A. Z. Msezane, editors.Contemporary problems in mathematical physics. World Scientific Publishing Co., Inc., River Edge, NJ, 2002

2002
[25]

E. Hille. Contributions to the theory of Hermitian series.Duke Math. J., 5:875–936, 1939

1939
[26]

E. Hille. Contributions to the theory of Hermitian series. II. The representation problem. Trans. Amer. Math. Soc., 47:80–94, 1940. RECURRENCE OPERATORS FOR GENERALIZED FOURIER SERIES 51

1940
[27]

E. Hille. Contributions to the theory of Hermitian series. III. Mean values.Internat. J. Math. Math. Sci., 3(3):407–421, 1980

1980
[28]

Hodges.A shorter model theory

W. Hodges.A shorter model theory. Cambridge University Press, Cambridge, 1997

1997
[29]

M. N. Hounkonnou, S. Belmehdi, and A. Ronveaux. Linearization of arbitrary products of classical orthogonal polynomials.Appl. Math. (Warsaw), 27(2):187–196, 2000

2000
[30]

W. T. Howell. On products of Laguerre polynomials.The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 24(161):396–405, 1937

1937
[31]

E. L. Ince.Ordinary differential equations. Dover Publications, New York, 1956

1956
[32]

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

2005
[33]

Koekoek, P

R. Koekoek, P. A. Lesky, and R. F. Swarttouw.Hypergeometric orthogonal polynomials and theirq-analogues. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2010

2010
[34]

Koepf and E

W. Koepf and E. N. Chiadjeu. Algorithmic approach for formal Fourier series.Math. Comput. Sci., 9(3):365–389, 2015

2015
[35]

Koutschan

C. Koutschan. A fast approach to creative telescoping.Mathematics in Computer Science, 4:259–266, 2010

2010
[36]

Lang.Algebra, volume 211 ofGraduate Texts in Mathematics

S. Lang.Algebra, volume 211 ofGraduate Texts in Mathematics. Springer-Verlag, New York, third edition, 2002

2002
[37]

Lewanowicz

S. Lewanowicz. Construction of a recurrence relation of the lowest order for coefficients of the Gegenbauer series.Zastos. Mat., 15(3):345–396, 1976

1976
[38]

Lewanowicz

S. Lewanowicz. Construction of the lowest-order recurrence relation for the Jacobi coefficients. Zastos. Mat., 17(4):655–675, 1983

1983
[39]

Lewanowicz

S. Lewanowicz. Recurrence relations for the coefficients in Jacobi series solutions of linear differential equations.SIAM J. Math. Anal., 17(5):1037–1052, 1986

1986
[40]

Lewanowicz

S. Lewanowicz. A new approach to the problem of constructing recurrence relations for the Jacobi coefficients.Zastos. Mat., 21(2):303–326, 1991

1991
[41]

Lewanowicz

S. Lewanowicz. Quick construction of recurrence relations for the Jacobi coefficients.J. Com- put. Appl. Math., 43(3):355–372, 1992

1992
[42]

Lewanowicz

S. Lewanowicz. Results on the associated classical orthogonal polynomials.J. Comput. Appl. Math., 65(1-3):215–231, 1995

1995
[43]

Lewanowicz

S. Lewanowicz. Second-order recurrence relation for the linearization coefficients of the clas- sical orthogonal polynomials.J. Comput. Appl. Math., 69(1):159 – 170, 1996

1996
[44]

Lewanowicz

S. Lewanowicz. Recurrences for the coefficients of series expansions with respect to classical orthogonal polynomials.Appl. Math. (Warsaw), 29(1):97–116, 2002

2002
[45]

Lewanowicz

S. Lewanowicz. Construction of recurrences for the coefficients of expansions inq-classical orthogonal polynomials.J. Comput. Appl. Math., 153(1-2):295–309, 2003

2003
[46]

Lewanowicz, E

S. Lewanowicz, E. Godoy, I. Area, A. Ronveaux, and A. Zarzo. Recurrence relations for the coefficients of the Fourier series expansions with respect toq-classical orthogonal polynomials. Numer. Algorithms, 23(1):31–50, 2000

2000
[47]

Lewanowicz and P

S. Lewanowicz and P. Woźny. Recurrence relations for the coefficients in series expansions with respect to semi-classical orthogonal polynomials.Numer. Algorithms, 35(1):61–79, 2004

2004
[48]

Y. L. Luke. Expansion of the confluent hypergeometric function in series of Bessel functions. Math. Tables Aids Comput., 13:261–271, 1959

1959
[49]

Y. L. Luke.The special functions and their approximations, Vol. I. Mathematics in Science and Engineering, Vol. 53. Academic Press, New York, 1969

1969
[50]

Y. L. Luke and R. L. Coleman. Expansion of hypergeometric functions in series of other hypergeometric functions.Math. Comp., 15:233–237, 1961

1961
[51]

Maroni and Z

P. Maroni and Z. da Rocha. Connection coefficients between orthogonal polynomials and the canonical sequence: an approach based on symbolic computation.Numer. Algorithms, 47(3):291–314, 2008

2008
[52]

Maroni and Z

P. Maroni and Z. da Rocha. Connection coefficients for orthogonal polynomials: symbolic computations, verifications and demonstrations in the Mathematica language.Numer. Algo- rithms, 63(3):507–520, 2013

2013
[53]

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

2003
[54]

Mora.Solving polynomial equation systems

T. Mora.Solving polynomial equation systems. Vol. IV. Buchberger theory and beyond, vol- ume 158 ofEncyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2016. 52 A. BENOIT, N. BRISEBARRE, AND B. SAL VY

2016
[55]

A. F. Nikiforov and V. B. Uvarov.Special functions of mathematical physics. Birkhäuser Verlag, Basel, 1988

1988
[56]

F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, editors.NIST Handbook of Mathematical Functions. Cambridge University Press, 2010

2010
[57]

Olver and A

S. Olver and A. Townsend. A fast and well-conditioned spectral method.SIAM Rev., 55(3):462–489, 2013

2013
[58]

O. Ore. Linear equations in non-commutative fields.Ann. of Math. (2), 32:463–477, 1931

1931
[59]

O. Ore. Theory of non-commutative polynomials.Ann. of Math. (2), 34(3):480–508, 1933

1933
[60]

Paszkowski.Zastosowania numeryczne wielomianów i szeregów Czebyszewa

S. Paszkowski.Zastosowania numeryczne wielomianów i szeregów Czebyszewa. Państwowe Wydawnictwo Naukowe, Warsaw, 1975. Podstawowe Algorytmy Numeryczne. [Fundamental Numerical Algorithms]

1975
[61]

Petkovšek

M. Petkovšek. Hypergeometric solutions of linear recurrences with polynomial coefficients.J. Symbolic Comput., 14(2-3):243–264, 1992

1992
[62]

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev.Integrals and Series. Volume 2: Special functions. Gordon and Breach, 1986

1986
[63]

A. P. Prudnikov, Y. A. Brychkov, and O. I. Marichev.Integrals and Series. Volume 3: More special functions. Gordon and Breach, 1989

1989
[64]

E. D. Rainville.Special functions. Chelsea Publishing Co., Bronx, N.Y., first edition, 1971

1971
[65]

Rebillard

L. Rebillard. Rational approximation in the complex plane using aτ-method and computer algebra.Numer. Algorithms, 16(2):187–208, 1997

1997
[66]

Rebillard.Étude théorique et algorithmique des séries de Chebyshev, solutions d’équations différentielles holonomes

L. Rebillard.Étude théorique et algorithmique des séries de Chebyshev, solutions d’équations différentielles holonomes. PhD thesis, Institut national polytechnique de Grenoble, 1998. https://theses.hal.science/tel-00008571/

1998
[67]

Rebillard and H

L. Rebillard and H. Zakrajšek. Recurrence relations for the coefficients in hypergeometric series expansions. In I. Kotsireas and E. Zima, editors,Computer Algebra 2006. Latest Ad- vances in Symbolic Algorithms, pages 158–180. World Sci. Publ., Hackensack, NJ, 2007

2006
[68]

Ronveaux

A. Ronveaux. Fourth-order differential equations for numerator polynomials.J. Phys. A, 21(15):L749–L753, 1988

1988
[69]

Ronveaux, M

A. Ronveaux, M. N. Hounkonnou, and S. Belmehdi. Generalized linearization problems.J. Phys. A, 28(15):4423–4430, 1995

1995
[70]

Ronveaux, A

A. Ronveaux, A. Zarzo, I. Area, and E. Godoy. Expansion of hypergeometric polynomials: several approaches. In Govaerts et al. [25], pages 446–459
[71]

Ronveaux, A

A. Ronveaux, A. Zarzo, and E. Godoy. Recurrence relations for connection coefficients be- tween two families of orthogonal polynomials.J. Comput. Appl. Math., 62(1):67–73, 1995

1995
[72]

Salvy and P

B. Salvy and P. Zimmermann. Gfun: a Maple package for the manipulation of generating and holonomic functions in one variable.ACM Trans. Math. Softw., 20(2):163–177, 1994

1994
[73]

P. Samuel. About Euclidean rings.J. Algebra, 19:282–301, 1971

1971
[74]

R. P. Stanley.Enumerative combinatorics, volume 2. Cambridge University Press, 1999

1999
[75]

Szegő.Orthogonal polynomials

G. Szegő.Orthogonal polynomials. American Mathematical Society, Providence, R.I., fourth edition, 1975. American Mathematical Society, Colloquium Publications, Vol. XXIII

1975
[76]

R. Szwarc. Linearization and connection coefficients of orthogonal polynomials.Monatsh. Math., 113(4):319–329, 1992

1992
[77]

H. C. Thacher, Jr. Conversion of a power to a series of Chebyshev polynomials.Comm. ACM, 7(3):181–182, 1964

1964
[78]

van der Put and M

M. van der Put and M. F. Singer.Galois theory of difference equations, volume 1666 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1997

1997
[79]

A. Verma. Certain expansions of the basic hypergeometric functions.Math. Comp., 20:151– 157, 1966

1966
[80]

Wimp.Computation with Recurrence Relations

J. Wimp.Computation with Recurrence Relations. Pitman, Boston, 1984

1984

Showing first 80 references.