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Revista de Matemática Teoría y Aplicaciones
Print version ISSN 1409-2433
Rev. Mat vol.18 n.1 San José Jun. 2011
A non-standard generating function for continuous dual q-hahn polynomials
Una función generatriz no estándar para polinomios q-hahn duales continuos
Una función generatriz no estándar para polinomios q-hahn duales continuos
*Facultad de Ciencias, Universidad Autónoma del Estado de Morelos, C.P. 62250 Cuernavaca, Morelos, México. E-Mail: mesuma@servm.fc.uaem.mx
†Instituto de Matemáticas, Unidad Cuernavaca, Universidad Nacional Autónoma de México, C.P. 62251 Cuernavaca, Morelos, México. E-Mail: natig@matcuer.unam.mx
Dirección para correspondencia
Abstract
We study a non-standard form of generating function for the three-parameter continuous dual q-Hahn polynomials p n(x; a, b, c | q), which has surfaced in a recent work of the present authors on the construction of lifting q-difference operators in the Askey scheme of basic hypergeometric polynomials. We show that the resulting generating function identity for the continuous dual q-Hahn polynomials p n(x; a, b, c | q) can be explicitly stated in terms of Jackson’s q-exponential functions eq(z).
Keywords: q-scheme of Askey, generating function, q-exponential function of Jackson, dual q-Hahn polynomials.
Resumen
Estudiamos una forma no estándar de la función generatriz para una familia de polinomios duales continuos q-Hahn de tres parámetros p n( x; a, b, c | q ), que han surgido en un trabajo reciente de los autores en la construcción de operadores elevadores en q-diferencias del esquema de Askey de polinomios básicos hipergeométricos. Demostramos que la función generatriz identidad resultante para los polinomios q-Hahn duales continuos p n(x; a, b, c | q) puede ser expresada explícitamente en términos de las funciones q-exponenciales de Jackson eq(z).
Palabras clave: esquema q de Askey, función generatriz, polinomios duales q-Hahn, función q-exponencial de Jackson.
Mathematics Subject Classification: 33D45, 39A70, 47B39.
Ver contenido disponible en pdf
We study a non-standard form of generating function for the three-parameter continuous dual q-Hahn polynomials p n(x; a, b, c | q), which has surfaced in a recent work of the present authors on the construction of lifting q-difference operators in the Askey scheme of basic hypergeometric polynomials. We show that the resulting generating function identity for the continuous dual q-Hahn polynomials p n(x; a, b, c | q) can be explicitly stated in terms of Jackson’s q-exponential functions eq(z).
Keywords: q-scheme of Askey, generating function, q-exponential function of Jackson, dual q-Hahn polynomials.
Resumen
Estudiamos una forma no estándar de la función generatriz para una familia de polinomios duales continuos q-Hahn de tres parámetros p n( x; a, b, c | q ), que han surgido en un trabajo reciente de los autores en la construcción de operadores elevadores en q-diferencias del esquema de Askey de polinomios básicos hipergeométricos. Demostramos que la función generatriz identidad resultante para los polinomios q-Hahn duales continuos p n(x; a, b, c | q) puede ser expresada explícitamente en términos de las funciones q-exponenciales de Jackson eq(z).
Palabras clave: esquema q de Askey, función generatriz, polinomios duales q-Hahn, función q-exponencial de Jackson.
Mathematics Subject Classification: 33D45, 39A70, 47B39.
Ver contenido disponible en pdf
References
[1] Wilf, H. (2004) Generatingfunctionology. Cambridge University Press, Cambridge. [ Links ]
[2] Hardy, G. (1999) Ramanujan: Twelwe Lectures on Subjects Suggested by His Life and Work. Chelsea, New York. [ Links ]
[3] Koekoek, R.; Swarttouw, R. (1998) “The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue”, Report 98–17, Delft University of Technology, Delft. [ Links ]
[4] Atakishiyeva, M.; Atakishiyev, N. (2010) “On lifting q-difference operators in the Askey scheme of basic hypergeometric polynomials”, J. Phys. A: Math. Theor. 43(14): 145201–145218. [ Links ]
[5] Exton H. (1983) q-Hypergeomteric Functions and Applications. Ellis Horwood, Chichester. [ Links ]
[6] Atakishiyev, N.; Suslov, S. (1992) “Difference Hypergeometric Functions”, in: A. Gonchar & E. Saff (Eds.) Progress in Approximation Theory: An International Perspective, Springer-Verlag, New York, Berlin: 1–35. [ Links ]
[7] Nelson, C.; Gartley, M. (1994) “On the zeros of the q-analogue exponential function”, J. Phys. A: Math. Gen. 27(11): 3857–3881. [ Links ]
[8] Rahman, M. (1995)“The q-exponential functions, old and new”, in: A. Sissakian & G. Pogosyan (Eds.) Proceedings of the International Workshop on Finite Dimensional Integrable Systems, Joint Institute for Nuclear Research, Dubna, Russia: 161–170. [ Links ]
[9] Atakishiyev, N. (1996) “On a one-parameter family of q-exponential functions”, J. Phys. A: Math. Gen. 29(10): L223–L227. [ Links ]
[10] Askey, R.; Wilson, J. (1985) “Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials”, Mem. Am. Math.Soc. 54(319): 1–55. [ Links ]
[11] Gasper, G.; Rahman, M. (2004) Basic Hypergeometric Functions. Cambridge University Press, Cambridge. [ Links ]
[12] Andrews, G.; Askey, R.; Roy, R. (1999) Special Functions. Cambridge University Press, Cambridge. [ Links ]
[13] Ismail, M. (2005) Classical and Quantum Orthogonal Polynomials in One Variable. Cambridge University Press, Cambridge. [ Links ]
[14] Atakishiyev, N. (1997) “Fourier–Gauss transforms of the Askey–Wilson polynomials”, J. Phys. A: Math. Gen. 30(24): L815–L820. [ Links ]
[1] Wilf, H. (2004) Generatingfunctionology. Cambridge University Press, Cambridge. [ Links ]
[2] Hardy, G. (1999) Ramanujan: Twelwe Lectures on Subjects Suggested by His Life and Work. Chelsea, New York. [ Links ]
[3] Koekoek, R.; Swarttouw, R. (1998) “The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue”, Report 98–17, Delft University of Technology, Delft. [ Links ]
[4] Atakishiyeva, M.; Atakishiyev, N. (2010) “On lifting q-difference operators in the Askey scheme of basic hypergeometric polynomials”, J. Phys. A: Math. Theor. 43(14): 145201–145218. [ Links ]
[5] Exton H. (1983) q-Hypergeomteric Functions and Applications. Ellis Horwood, Chichester. [ Links ]
[6] Atakishiyev, N.; Suslov, S. (1992) “Difference Hypergeometric Functions”, in: A. Gonchar & E. Saff (Eds.) Progress in Approximation Theory: An International Perspective, Springer-Verlag, New York, Berlin: 1–35. [ Links ]
[7] Nelson, C.; Gartley, M. (1994) “On the zeros of the q-analogue exponential function”, J. Phys. A: Math. Gen. 27(11): 3857–3881. [ Links ]
[8] Rahman, M. (1995)“The q-exponential functions, old and new”, in: A. Sissakian & G. Pogosyan (Eds.) Proceedings of the International Workshop on Finite Dimensional Integrable Systems, Joint Institute for Nuclear Research, Dubna, Russia: 161–170. [ Links ]
[9] Atakishiyev, N. (1996) “On a one-parameter family of q-exponential functions”, J. Phys. A: Math. Gen. 29(10): L223–L227. [ Links ]
[10] Askey, R.; Wilson, J. (1985) “Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials”, Mem. Am. Math.Soc. 54(319): 1–55. [ Links ]
[11] Gasper, G.; Rahman, M. (2004) Basic Hypergeometric Functions. Cambridge University Press, Cambridge. [ Links ]
[12] Andrews, G.; Askey, R.; Roy, R. (1999) Special Functions. Cambridge University Press, Cambridge. [ Links ]
[13] Ismail, M. (2005) Classical and Quantum Orthogonal Polynomials in One Variable. Cambridge University Press, Cambridge. [ Links ]
[14] Atakishiyev, N. (1997) “Fourier–Gauss transforms of the Askey–Wilson polynomials”, J. Phys. A: Math. Gen. 30(24): L815–L820. [ Links ]
Correspondencia a: Mesuma Atakishiyeva. Facultad de Ciencias, Universidad Autónoma del Estado de Morelos, C.P. 62250 Cuernavaca, Morelos, México. E-Mail: mesuma@servm.fc.uaem.mx
Natig Atakishiyev. Instituto de Matemáticas, Unidad Cuernavaca, Universidad Nacional Autónoma de México, C.P. 62251 Cuernavaca, Morelos, México. E-Mail: natig@matcuer.unam.mx
Natig Atakishiyev. Instituto de Matemáticas, Unidad Cuernavaca, Universidad Nacional Autónoma de México, C.P. 62251 Cuernavaca, Morelos, México. E-Mail: natig@matcuer.unam.mx
Received: 18 Feb 2010; Revised: 22 Oct 2010; Accepted: 23 Nov 2010
