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Cohl, Howard S.; Costas-Santos, Roberto S.; Wakhare, Tanay V.

Some generating functions for \(q\)-polynomials. (English) Zbl 1425.33009

Symmetry 10, No. 12, Paper No. 758, 12 p. (2018).
Summary: Demonstrating the striking symmetry between calculus and \(q\)-calculus, we obtain \(q\)-analogues of the Bateman, Pasternack, Sylvester, and Cesàro polynomials. Using these, we also obtain \(q\)-analogues for some of their generating functions. Our \(q\)-generating functions are given in terms of the basic hypergeometric series \(_4\phi_5\), \(_5 \phi_5\), \(_4 \phi_3\), \(_3 \phi_2\), \(_2 \phi_1\), and \(q\)-Pochhammer symbols. Starting with our \(q\)-generating functions, we are also able to find some new classical generating functions for the Pasternack and Bateman polynomials.

Cited in 1 Document

MSC:

33D45 Basic orthogonal polynomials and functions (Askey-Wilson polynomials, etc.)
33C20 Generalized hypergeometric series, \({}_pF_q\)

Keywords:

basic hypergeometric functions; generating functions; \(q\)-polynomials
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\textit{H. S. Cohl} et al., Symmetry 10, No. 12, Paper No. 758, 12 p. (2018; Zbl 1425.33009)
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References:

[1] Koekoek, R.; Lesky, P.A.; Swarttouw, R.F.; ; Hypergeometric Orthogonal Polynomials and Their q-Analogues: Berlin, Germany 2010; . · Zbl 1200.33012
[2] Gasper, G.; Rahman, M.; ; Basic Hypergeometric Series: Cambridge, UK 2004; . · Zbl 1129.33005
[3] Bateman, H.; Two systems of polynomials for the solution of Laplace’s integral equation; Duke Math. J.: 1936; Volume 2 ,569-577. · Zbl 0015.11502
[4] Sylvester, J.J.; Sur la valeur moyenne des coefficients dans le développement d’un déterminant gauche ou symétrique d’un ordre infiniment grand et sur les déterminants doublement gauches; C. R. de l’Académie des Sci.: 1879; Volume 89 ,24-26. · JFM 11.0110.02
[5] Erdélyi, A.; Magnus, W.; Oberhettinger, F.; Tricomi, F.G.; ; Higher Transcendental Functions: Melbourne, Australia 1981; . · Zbl 0051.30303
[6] Srivastava, H.M.; Manocha, H.L.; ; A Treatise on Generating Functions: Chichester, UK 1984; ,569. · Zbl 0535.33001
[7] Rainville, E.D.; ; Special Functions: New York, NY, USA 1960; . · Zbl 0092.06503
[8] Bateman, H.; Some Properties of a certain Set of Polynomials; Tohoku Math. J. First Ser.: 1933; Volume 37 ,23-38. · Zbl 0007.30701
[9] Pasternack, S.; A generalization of the polynomial Fn(x); Lond. Edinb. Dublin Philos. Mag. J. Sci. Ser. 7: 1939; Volume 28 ,209-226. · JFM 65.1216.01
[10] Koelink, H.T.; On Jacobi and continuous Hahn polynomials; Proc. Am. Math. Soc.: 1996; Volume 124 ,887-898. · Zbl 0848.33007
[11] Agarwal, A.K.; Manocha, H.L.; On some new generating functions; Matematicki Vesnik: 1980; Volume 4 ,395-402. · Zbl 0514.33004
[12] Srivastava, H.M.; Lavoie, J.L.; Tremblay, R.; The Rodrigues Type Representations for a Certain Class of Special Functions; Annali di Matematica Pura ed Applicata: 1979; Volume 119 ,9-24. · Zbl 0386.33007
[13] Kim, D.S.; Kim, T.K.; q-Bernoulli polynomials and q-umbral calculus; Sci. China Math.: 2014; Volume 57 ,1867-1874. · Zbl 1303.05015
[14] Asif, M.; On Some Problems in Special Functions; Ph.D. Thesis: Aligarh, India 2010; .
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