Abstract
In this chapter, we present a rigorous development of I. M. Sheffer’s characterization of the A-Type 0 orthogonal polynomial sequences. We first develop the results that led to the main theorem that characterizes the general A-Type 0 polynomial sequences via a linear generating function. From there, we develop the additional theory that Sheffer utilized in order to determine which A-Type 0 polynomial sequences are also orthogonal. We then address Sheffer’s additional characterizations of B-Type and C-Type, as well as E.D. Rainville’s \(\sigma \) -Type classification. Lastly, we cover J. Meixner’s approach to the same characterization problem studied by Sheffer and then discuss an extension of Meixner’s analysis by W.A. Al-Salam. Portions of the analysis addressed throughout this chapter are supplemented with informative concrete examples.
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References
W.A. Al-Salam, On a Characterization of Meixner’s Polynomials, Quart. J. Math. Oxford, 17(1966), 7–10.
W.A. Al-Salam, Characterization theorems for orthogonal polynomials, in: Nevai, P. (Ed.), Orthogonal Polynomials: Theory and Practice. Kluwer Academic Publishers, Dordrecht, pp. 1–24, (1990).
G.E. Andrews, R.A. Askey and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications 71, Cambridge University Press, Cambridge, 1999.
G. Gasper and M. Rahman, Basic Hypergeometric Series, second ed., Encyclopedia of Mathematics and its Applications 96, Cambridge University Press, Cambridge, 2004.
M.E.H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Encyclopedia of Mathematics and its Applications 98, Cambridge University Press, Cambridge, 2005.
R. Koekoek and R.F. Swarttouw, The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Reports of the Faculty of Technical Mathematics and Information, No. 98–17, Delft University of Technology, (1998). http://aw.twi.tudelft.nl/~koekoek/askey/index.html
J. Meixner, Orthogonale Polynomsysteme mit einer besonderen Gestalt der erzeugenden Funktion, J. London Math. Soc., 9(1934), 6–13.
E.D. Rainville, Special Functions, Macmillan, New York, 1960.
I.M. Sheffer, Some properties of polynomial sets of type zero, Duke Math J., 5(1939), 590–622.
G. Szegő, Orthogonal Polynomials, fourth ed., American Mathematical Society, Colloquium Publications, Vol. XXIII, Providence, 1975.
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© 2013 Daniel J. Galiffa
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Galiffa, D.J. (2013). The Sheffer A-Type 0 Orthogonal Polynomial Sequences and Related Results. In: On the Higher-Order Sheffer Orthogonal Polynomial Sequences. SpringerBriefs in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5969-9_1
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DOI: https://doi.org/10.1007/978-1-4614-5969-9_1
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