On a power series involving classical orthogonal polynomials

Romanian Reports of Physics

Volumn 55, Issue 5-6, 2010, Pages 631-644

  Marian, Paulina ^a   Marian, Tudor A ^a  

^a UNIVERSITY OF BUCHAREST   (Romania)

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EID: 77955500327     PISSN: 1221146X     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (30)

1. Paulina Marian, Phys. Rev. A 45, 2044-2051 (1992), Appendix B.
2. Paulina Marian and Tudor A. Marian, Phys. Rev. A 47, 4487-4495 (1993), Appendix B.
3. Ref. [2], Appendix A and Section IV.
4. Earl D. Rainville, Special Functions (Macmillan, New York, 1960).
5. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions (McGraw-Hill, New York, 1953), Vol. 2, p. 177, eq. (29).
6. Lars V. Ahlfors, Complex Analysis, Third Edition, (McGraw-Hill, New York, 1979), pp. 114-123.
7. Einar Hille, Analytic Function Theory, Vol. 1, Second Edition (Chelsea, New York, 1982), pp. 175-182.
8. Ref. [4], p. 280, § 144, eq. (23).
9. Ref. [4], p. 169, § 95, eq. (7).
10. Ref. [5], Vol. 2, p. 184, eq. (5).
11. Ref. [5], Vol. 2, p. 174, eq. (6), where the limit should be read as limλ→0.
12. Ref. [5], Vol. 2, p. 186, eq. (30): there is a mistake that can be corrected by omitting unity in the l.h.s. of this equation.
13. See Ref. [5], Vol. 2, p. 186, eq. (29): our eq. (3.14) is equivalent to this equation.
14. Ref. [5], Vol. 2, p. 184, eq. (6).
15. Ref. [5], Vol. 2, p. 189, eq. (17).
16. (α)0(t,u) and coincides with eq. (4.3) of the present work.
17. Ref. [4], p. 211, §119, eq. (9).
18. Ref. [5], Vol. 2, p. 194, eq. (19).
19. Ref. [4], p.197, § 111, eq. (1).
20. George E. Andrews, Richard Askey, and Ranjan Roy, Special Functions (Cambridge University Press, Cambridge, UK, 2000), p. 306.
21. Ref. [20], p. 339.
22. Ref. [5], Vol. 2, p. 194, eq. (22).
23. Herbert Buchholz, The Confluent Hypergeometric Function (Springer, Berlin, 1969), p.147, eq. (12b). The original reference is: Ferdinand Gustav Mehler, Über die Entwicklung einer Funktion von beliebig vielen Variablen nach Laplaceschen Funktionen höherer Ordnung, J. reine angew. Math. 66, 161-176 (1866).
24. Ref. [20], pp. 280-282.
25. Ref. [23], p. 147, eq. (12β).
26. Ref. [5], Vol. 2, p. 193, eq. (2).
27. Ref. [20], pp. 348-350. See the generating functions of some nonclassical orthogonal polynomials listed there.
28. Eugen Merzbacher, Quantum Mechanics, Third Edition, (John Wiley, New York, 1998), p. 352, eq. (15.59).
29. R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path Integrals, (McGraw-Hill, New York, 1965), p. 198, eq. (8-1). Our eq. (7.8) coincides with their explicit expression of the propagator. In order to derive it, the authors of the book point out several intermediate steps to be achieved: p. 28, eq. (2-9); p. 63, eq. (3-59); p. 73, eq. (3-93).
30. Hagen Kleinert, Path Integrals in Quantum Mechanics, Statistics, and Polymer Physics, Second Edition, (World Scientific, Singapore, 1995), p. 93, eq. (2.148).