## Volterra-type convolution of classical polynomials

HTML articles powered by AMS MathViewer

- by
Ana F. Loureiro and Kuan Xu
**HTML**| PDF - Math. Comp.
**88**(2019), 2351-2381 Request permission

## Abstract:

We present a general framework for calculating the Volterra-type convolution of polynomials from an arbitrary polynomial sequence $\{P_k(x)\}_{k \geqslant 0}$ with $\deg P_k(x) = k$. Based on this framework, series representations for the convolutions of classical orthogonal polynomials, including Jacobi and Laguerre families, are derived, along with some relevant results pertaining to these new formulas.## References

- George E. Andrews, Richard Askey, and Ranjan Roy,
*Special functions*, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, Cambridge, 1999. MR**1688958**, DOI 10.1017/CBO9781107325937 - I. Area, E. Godoy, A. Ronveaux, and A. Zarzo,
*Solving connection and linearization problems within the Askey scheme and its $q$-analogue via inversion formulas*, Proceedings of the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Patras, 1999), 2001, pp.Â 151â€“162. MR**1858275**, DOI 10.1016/S0377-0427(00)00640-3 - Richard Askey,
*Orthogonal polynomials and special functions*, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975. MR**0481145** - Richard Askey and James Fitch,
*Integral representations for Jacobi polynomials and some applications*, J. Math. Anal. Appl.**26**(1969), 411â€“437. MR**237847**, DOI 10.1016/0022-247X(69)90165-6 - Kendall E. Atkinson,
*The numerical solution of integral equations of the second kind*, Cambridge Monographs on Applied and Computational Mathematics, vol. 4, Cambridge University Press, Cambridge, 1997. MR**1464941**, DOI 10.1017/CBO9780511626340 - H. Brunner and P. J. van der Houwen,
*The numerical solution of Volterra equations*, CWI Monographs, vol. 3, North-Holland Publishing Co., Amsterdam, 1986. MR**871871** - W. Bryc, W. Matysiak, R. Szwarc, and J. Wesolowski,
*Projection formulas for orthogonal polynomials*, arXiv:math/0606092v2, 2007. - Steven B. Damelin and Willard Miller Jr.,
*The mathematics of signal processing*, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2012. MR**2883645** - Dean G. Duffy,
*Greenâ€™s functions with applications*, 2nd ed., Advances in Applied Mathematics, CRC Press, Boca Raton, FL, 2015. MR**3379916**, DOI 10.1201/b18159 - D. Forsyth and J. Ponce,
*Computer Vision: A Modern Approach*, Prentice Hall, 2011. - Izrail Solomonovich Gradshteyn and Iosif Moiseevich Ryzhik,
*Table of integrals, series, and products*, Academic press, 2014. - Thomas Hagstrom,
*Radiation boundary conditions for the numerical simulation of waves*, Acta numerica, 1999, Acta Numer., vol. 8, Cambridge Univ. Press, Cambridge, 1999, pp.Â 47â€“106. MR**1819643**, DOI 10.1017/S0962492900002890 - R. Hilfer (ed.),
*Applications of fractional calculus in physics*, World Scientific Publishing Co., Inc., River Edge, NJ, 2000. MR**1890104**, DOI 10.1142/9789812817747 - Robert V. Hogg and Allen T. Craig,
*Introduction to mathematical statistics*, 3rd ed., The Macmillan Company, New York; Collier Macmillan Ltd., London, 1970. MR**0251823** - Mourad E. H. Ismail,
*Classical and quantum orthogonal polynomials in one variable*, Encyclopedia of Mathematics and its Applications, vol. 98, Cambridge University Press, Cambridge, 2009. With two chapters by Walter Van Assche; With a foreword by Richard A. Askey; Reprint of the 2005 original. MR**2542683** - Mourad E. H. Ismail, Erik Koelink, and Pablo RomĂˇn,
*Generalized Burchnall-type identities for orthogonal polynomials and expansions*, SIGMA Symmetry Integrability Geom. Methods Appl.**14**(2018), Paper No. 072, 24. MR**3828871**, DOI 10.3842/SIGMA.2018.072 - H. T. Koelink and J. Van Der Jeugt,
*Convolutions for orthogonal polynomials from Lie and quantum algebra representations*, SIAM J. Math. Anal.**29**(1998), no.Â 3, 794â€“822. MR**1617724**, DOI 10.1137/S003614109630673X - Wolfram Koepf and Dieter Schmersau,
*Representations of orthogonal polynomials*, J. Comput. Appl. Math.**90**(1998), no.Â 1, 57â€“94. MR**1627168**, DOI 10.1016/S0377-0427(98)00023-5 - S. Lewanowicz,
*The hypergeometric functions approach to the connection problem for the classical orthogonal polynomials*, Technical Report, 1998. - Peter Linz,
*Analytical and numerical methods for Volterra equations*, SIAM Studies in Applied Mathematics, vol. 7, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1985. MR**796318**, DOI 10.1137/1.9781611970852 - Yudell L. Luke,
*The special functions and their approximations, Vol. I*, Mathematics in Science and Engineering, Vol. 53, Academic Press, New York-London, 1969. MR**0241700** - P. Maroni,
*Semi-classical character and finite-type relations between polynomial sequences*, Appl. Numer. Math.**31**(1999), no.Â 3, 295â€“330. MR**1711161**, DOI 10.1016/S0168-9274(98)00137-8 - P. Maroni and Z. da Rocha,
*Connection coefficients between orthogonal polynomials and the canonical sequence: an approach based on symbolic computation*, Numer. Algorithms**47**(2008), no.Â 3, 291â€“314. MR**2385739**, DOI 10.1007/s11075-008-9184-9 - Frank W. J. Olver, Daniel W. Lozier, Ronald F. Boisvert, and Charles W. Clark (eds.),
*NIST handbook of mathematical functions*, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010. With 1 CD-ROM (Windows, Macintosh and UNIX). MR**2723248** - Earl D. Rainville,
*Special functions*, 1st ed., Chelsea Publishing Co., Bronx, N.Y., 1971. MR**0393590** - J. SĂˇnchez-Ruiz, P. L. ArtĂ©s, A. MartĂnez-Finkelshtein, and J. S. Dehesa,
*General linearization formulae for products of continuous hypergeometric-type polynomials*, J. Phys. A**32**(1999), no.Â 42, 7345â€“7366. MR**1747172**, DOI 10.1088/0305-4470/32/42/308 - Eduardo D. Sontag,
*Mathematical control theory*, 2nd ed., Texts in Applied Mathematics, vol. 6, Springer-Verlag, New York, 1998. Deterministic finite-dimensional systems. MR**1640001**, DOI 10.1007/978-1-4612-0577-7 - Ivar Stakgold and Michael Holst,
*Greenâ€™s functions and boundary value problems*, 3rd ed., Pure and Applied Mathematics (Hoboken), John Wiley & Sons, Inc., Hoboken, NJ, 2011. MR**2789179**, DOI 10.1002/9780470906538 - Gabor SzegĂ¶,
*Orthogonal Polynomials*, American Mathematical Society Colloquium Publications, Vol. 23, American Mathematical Society, New York, 1939. MR**0000077** - D. D. Tcheutia,
*On connection, linearization and duplication coefficients of classical orthogonal polynomials*, Ph.D. thesis, Kassel: UniversitĂ¤tsbibliothek, 2014. - Lloyd N. Trefethen,
*Approximation theory and approximation practice*, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2013. MR**3012510** - C. Wagner, T. HĂĽttl, and P. Sagaut,
*Large-Eddy Simulation for Acoustics*, Cambridge University Press, 2007. - Jet Wimp,
*Connection coefficients, orthogonal polynomials and the WZ-algorithms*, Numer. Algorithms**21**(1999), no.Â 1-4, 377â€“386. Numerical methods for partial differential equations (Marrakech, 1998). MR**1725736**, DOI 10.1023/A:1019117731699 - Kuan Xu and Ana F. Loureiro,
*Spectral approximation of convolution operators*, SIAM J. Sci. Comput.**40**(2018), no.Â 4, A2336â€“A2355. MR**3835595**, DOI 10.1137/17M1149249 - A. Zarzo, I. Area, E. Godoy, and A. Ronveaux,
*Results for some inversion problems for classical continuous and discrete orthogonal polynomials*, J. Phys. A**30**(1997), no.Â 3, L35â€“L40. MR**1449238**, DOI 10.1088/0305-4470/30/3/002

## Additional Information

**Ana F. Loureiro**- Affiliation: School of Mathematics, Statistics and Actuarial Sciences, University of Kent, Sibson Building, Canterbury, CT2 7FS, United Kingdom
- MR Author ID: 793430
- Email: A.Loureiro@kent.ac.uk
**Kuan Xu**- Affiliation: School of Mathematical Sciences, University of Science and Technology of China, 96 Jinzhai Road, Hefei, Anhui, 230026, Peopleâ€™s Republic of China
- MR Author ID: 1010592
- Email: kuanxu@ustc.edu.cn
- Received by editor(s): April 26, 2018
- Received by editor(s) in revised form: October 29, 2018
- Published electronically: April 2, 2019
- Additional Notes: The first author is the corresponding author

The second author was funded in part by Anhui Initiative in Quantum Information Technologies under grant AHY150200. - © Copyright 2019 American Mathematical Society
- Journal: Math. Comp.
**88**(2019), 2351-2381 - MSC (2010): Primary 42A85, 44A35, 33C05, 33C20, 33C45; Secondary 42A10, 41A10, 41A58
- DOI: https://doi.org/10.1090/mcom/3427
- MathSciNet review: 3957896