Abstract
A new constructive approach is given to the linearization formulas of symmetric orthogonal polynomials. We use the monic three-term recurrence relation of an orthogonal polynomial system to set up a partial difference equation problem for the product of two polynomials and solve it in terms of the initial data. To this end, an auxiliary function of four integer variables is introduced, which may be seen as a discrete analogue of Riemann's function. As an application, we derive the linearization formulas for the associated Hermite polynomials and for their continuousq-analogues. The linearization coefficients are represented here in terms of3 F 2 and3Φ2 (basic) hypergeometric functions, respectively. We also give some partial results in the case of the associated continuousq-ultraspherical polynomials.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Askey (1970):Linearization of the product of orthogonal polynomials. In: Problems in Analysis (R. Gunning, ed.), Princeton, NJ: Princeton University Press, pp. 131–138.
R. Askey (1975): Orthogonal Polynomials and Special Functions. Philadelphia, PA: SIAM.
R. Askey, M. E. H. Ismail (1985): Recurrence Relations, Continuous Fractions and Orthogonal Polynomials. Memoirs of the American Mathematical Society, number 300. Providence, RI: American Mathematical Society.
J. Bustoz, M. E. H. Ismail (1982):The associated ultraspherical polynomials and their q-analogues. Canad. J. Math.,34:718–736.
T. S. Chihara (1978): An Introduction to Orthogonal Polynomials. New York: Gordon and Breach.
W. C. Connett, C. Markett, A. L. Schwartz (1992):Convolution and hypergroup structures associated with a class of Sturm-Liouville expansions. Trans. Amer. Math. Soc.332:365–390.
W. C. Connett, C. Markett, A. L. Schwartz (1992):Jacobi polynomials and related hypergroup structures. In: Probability Measures on Groups (H. Heyer, ed.). New York: Plenum, pp. 45–81.
A. Erdélyi, W. Magnus, F. Oberhettinger, F. G. Tricomi (1953): Higher Transcendental Functions, vol. I. New York: McGraw-Hill.
G. Gasper (1970):Linearization of the product of Jacobi polynomials, I, II. Canad. J. Math.,22:171–175, 582–593.
G. Gasper (1975)Positivity and special functions. In: Theory and Application of Special Functions (R. Askey, ed.). New York: Academic Press, pp. 375–433.
G. Gasper (1983):A convolution structure and positivity of a generalized translation operator for the continuous q-Jacobi polynomials. Proceedings of a Conference in Harmonic Analysis in Honour of A. Zygmund (W. Beckner et al., eds.). Belmont, CA: Wadsworth, pp. 44–59.
G. Gasper, M. Rahman (1990): Basic Hypergeometric Series. Cambridge: Cambridge University Press.
H. Heyer (1990):Convolution semigroups and potential kernels on a commutative hypergroup. In: The Analytical and Topological Theory of Semigroups (K. H. Hofmann, J. D. Lawson, J. S. Pym, eds.). New York: de Gruyter, pp. 279–312.
I. I. Hirschman, Jr. (1956):Harmonic analysis and ultraspherical polynomials. Symposium on Harmonic Analysis and Related Integral Transforms, Cornell University.
M. Kracht, E. Kreyszig (1988): Methods of Complex Analysis in Partial Differential Equations with Applications. New York: Wiley.
R. Lasser (1983):Linearization of the product of associated Legendre polynomials. SIAM J. Math. Anal.,14:403–408.
R. Lasser (1983):Orthogonal polynomials and hypergroups. Rend. mat.,3:185–209.
R. Lasser: (1992):Orthogonal polynomials and hypergroups, II—the symmetric case. Preprint, 38 pp.
C. Markett (1984):Product formulas for Bessel, Whittaker, and Jacobi functions via the solution of an associated Cauchy problem. In.: Functional Analysis and Approximation (P. L. Butzer, R. L. Stens, B. Sz.-Nagy, eds.). International Series of Numerical Mathematics, vol. 65. Basel: Birkhäuser, pp. 449–462.
C. Markett (1984):Norm estimates for generalized translation operators associated with a singular differential operator. Nederl. Akad. Wetensch. Indag. Math.,46:299–313.
C. Markett (1985): Produktformeln für Eigenfunktionen singulärer Sturm-Liouville Gleichungen und verallgemeinerte Translationsoperatoren. Habilitationsschrift, RWTH Aachen, 132 pp.
C. Markett (1989):Product formulas and convolution structure for Fourier-Bessel series. Constr. Approx.,5:383–404.
C. Markett, J. Püngel, H. Wallner (1990):Multiple power series representations for Riemann functions of self-adjoint equations. Applicable Anal.,38:179–199.
M. Rahman (1981):A non-negative representation of the linearization coefficients of the product of Jacobi polynomials. Canad. J. Math.,33:915–928.
M. Rahman (1981):The linearization of the product of continuous q-Jacobi polynomials. Canad. J. Math.,33:961–987.
L. J. Slater (1966): Generalized Hypergeometric Functions. Cambridge: Cambridge University Press.
R. Szwarc (preprint):Orthogonal polynomials and a discrete boundary value problem, I. SIAM J. Math. Anal.
R. Szwarc (preprint):Orthogonal polynomials and a discrete boundary value problem, II. SIAM J. Math. Anal.
R. Szwarc (preprint):Convolution structures associated with orthogonal polynomials. J. Math. Anal. Appl.
M. Voit (1990):Laws of large numbers for polynomial hypergroups and some applications. J. Theoret. Probab.,3:245–266.
M. Voit (1990):Central limit theorems for a class of polynomial hypergroups. Adv. in Appl. Probab.,22:66–87.
Author information
Authors and Affiliations
Additional information
Communicated by Mourad Ismail.
Rights and permissions
About this article
Cite this article
Markett, C. Linearization of the product of symmetric orthogonal polynomials. Constr. Approx 10, 317–338 (1994). https://doi.org/10.1007/BF01212564
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01212564