A basic class of symmetric orthogonal polynomials using the extended Sturm-Liouville theorem for symmetric functions. (English) Zbl 1131.33005

Using the extended Sturm-Liouville theorem a basic class of symmetric orthogonal polynomials with four free parameters is introduced. All the standard properties are obtained, including the explicit form, a generic second order differential equation, a generic orthogonality relation and a generic three term recurrence relation. Next it is shown that four classes of symmetric polynomials (generalized ultraspherical polynomials, generalized Hermite polynomials, and two finite classes of symmetric polynomials which are finitely orthogonal on \((-\infty, \infty)\) with respect to special weight functions) can be obtained from the introduced class and consequently all its standard properties can be generated. Two kinds of half-trigonometric orthogonal polynomials appears that the author calls fifth and sixth kind Chebyshev polynomials because they are generated using the first and second kind Chebyshev polynomials and have the half-trigonometric forms.


33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
33C47 Other special orthogonal polynomials and functions
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
Full Text: DOI arXiv


[1] Abramowitz, M.; Stegun, I.A., Handbook of mathematical functions with formulas, graphs, and mathematical tables, (1972), Dover New York · Zbl 0543.33001
[2] Al-Salam, W.; Allaway, W.R.; Askey, R., Sieved ultraspherical polynomials, Trans. amer. math. soc., 284, 39-55, (1984) · Zbl 0547.33005
[3] Arfken, G., Mathematical methods for physicists, (1985), Academic Press · Zbl 0135.42304
[4] Askey, R., Orthogonal polynomials old and new, and some combinatorial connections, (), 67-84
[5] Charris, J.A.; Ismail, M.E.H., On sieved orthogonal polynomials II: random walk polynomials, Canad. J. math., 38, 397-415, (1986) · Zbl 0576.33006
[6] Chihara, T.S., Introduction to orthogonal polynomials, (1978), Gordon & Breach New York · Zbl 0389.33008
[7] Cover, T.M.; Thomas, J.A., Elements of information theory, (1991), Wiley-Interscience New York · Zbl 0762.94001
[8] Dette, H., Characterizations of generalized Hermite and sieved ultraspherical polynomials, Proc. amer. math. soc., 384, 691-711, (1996) · Zbl 0863.33006
[9] Fisher, R.A., Theory of statistical estimation, Proc. Cambridge philos. soc., 22, 700-725, (1925) · JFM 51.0385.01
[10] Fox, L.; Parker, I.B., Chebyshev polynomials in numerical analysis, (1968), Oxford Univ. Press London, England · Zbl 0153.17502
[11] Ismail, M.E.H., On sieved orthogonal polynomials III: orthogonality on several intervals, Trans. amer. math. soc., 294, 89-111, (1986) · Zbl 0612.33009
[12] Koepf, W., Hypergeometric summation, (1988), Vieweg Braunschweig/Wiesbaden
[13] Konoplev, V.P., The asymptotic behavior of orthogonal polynomials at one-sided singular points of weighting functions (algebraic singularities), Soviet math. dokl., 6, 223-227, (1965) · Zbl 0132.29604
[14] Masjed-Jamei, M., Three finite classes of hypergeometric orthogonal polynomials and their application in functions approximation, Integral transforms spec. funct., 13, 169-190, (2002) · Zbl 1017.33005
[15] Masjed-Jamei, M., Classical orthogonal polynomials with weight function \(((a x + b)^2 +(c x + d)^2)^{- p} \exp(q \arctan \frac{a x + b}{c x + d})\); \(x \in(- \infty, \infty)\) and a generalization of T and F distributions, Integral transforms spec. funct., 15, 137-153, (2004) · Zbl 1055.33009
[16] M. Masjed-Jamei, A generalization of classical symmetric orthogonal functions using a symmetric generalization of Sturm-Liouville problems, J. Integral Transforms and Special Functions, in press · Zbl 1133.34020
[17] M. Masjed-Jamei, A basic class of discrete orthogonal polynomials using the extended Sturm-Liouville theorem in discrete spaces, submitted for publication · Zbl 1131.33005
[18] Nevai, P., Orthogonal polynomials, Mem. amer. math. soc., vol. 213, (1979), Amer. Math. Soc. Providence, RI · Zbl 0405.33009
[19] Nikiforov, A.F.; Uvarov, V.B., Special functions of mathematical physics, (1988), Birkhäuser Basel · Zbl 0694.33005
[20] Rivlin, T., Chebyshev polynomials: from approximation theory to algebra and number theory, (1990), Wiley New York · Zbl 0734.41029
[21] Rogers, L.J., Third memoir on the expansion of certain infinite products, Proc. London math. soc., 26, 15-32, (1895) · JFM 26.0289.01
[22] Szegö, G., Orthogonal polynomials, Amer. math. soc. colloq. publ., vol. 23, (1975), Providence RI · JFM 65.0278.03
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.