On generating functions of biorthogonal polynomials suggested by the Laguerre polynomials
DOI:
https://doi.org/10.4067/S0716-09172014000200001Keywords:
Laguerre polynomials, biorthogonal polynomials, generating functions, polinomios de Laguerre, polinomios biortogonales, funciones generatrices.Abstract
In this note, we have obtained some novel bilateral generating functions involving Konhauser biorthogonal polynomials, which is converted into trilateral generating functions with Tchebycheff polynomials by group theoretic method. As special cases, we have obtained the corresponding results on generalised Laguerre polynomials. Some applications are also given here.References
[1] Konhauser, J. D. E., Some properties of biorthogonal polynomials, J. Math. Anal. Appl., 11 , pp. 242-260, (1965).
[2] Konhauser, J. D. E., Biorthogonal polynomials suggested by the Laguerre polynomials, Pacific J. Math., 21, pp. 303-314, (1967).
[3] Spencer, L. and Fano, U., Penetration and diffusion of X-rays , calculation of spatial distribution by polynomial expansion, J. Res. Nat. Bur. Standards 46, pp. 446-461, (1951).
[4] Erdelyi, Arthur, with Magnus, W., Oberhettinger, F., Tricomi, F. G., et al., Higher transcendental function, vol. 2. New York: McGraw-Hill, (1953).
[5] Carlitz, L., A note on certain biorthogonal polynomials, Pacific J. Math., 24, pp. 425-430 (1968).
[6] Carlitz, L., A note on the Laguerre polynomials, Michigan Math. J., 7(3), pp. 219-223, (1960).
[7] Al-Salam, W. A., Operational representations for the Laguerre and other polynomials, Duke Math. Jour., 31, pp. 127-142, (1964).
[8] Calvez, L. C. et Genin, R., Applications des relations entre les fonctions generatrices et les formules de type Rodrigues, C. R. Acad. Sci. Paris Ser. A-B, 270, pp. A41-A44, (1970).
[9] Andrews,L.C., Special for Engineers and Applied Mathematicians, Macmillan Publishing Company.
[10] Prabhakar, T. R., On a set of polynomials suggested by the Laguerre polynomials, Pacific J. Math., 35 , pp. 213-219, (1970).
[11] Prabhakar, T. R., On the other set of biorthogonal polynomials suggested by Laguerre polynomials, Pacific J. Math., 37, pp. 801-804, (1971).
[12] Srivastava, H. M., Some Biorthogonal polynomials suggested by the Laguerre polynomials, Pacific J. Math., 98, pp. 235-250, (1982).
[13] Srivastava, H. M., A note on the Konhauser sets of Biorthogonal polynomials suggested by the Laguerre polynomials, Pacific J. Math., 91, pp. 235-250, (1980).
[14] Srivastava, A. N. and Singh, S. N., On the Konhauser polynomials Yna (x; k), Indian J. pure appl. Math., 10, pp. 1121-1126, (1979).
[15] Shreshtha, R.M. and Bajracharya, S., Group theoretical study of a bilateral generating function, Int. Trans. and Spec. func., 5(1-2), pp. 147-152, (1997), 147-152.
[16] Truesdell, C., An essay toward a unified theory of special functions, Princeton university press, (1948).
[17] McBride, E. B., Obtaining Generating Functions, Springer Verlag, Berlin, (1972).
[18] Sharma, R. and Chongdar, A. K., Some generating functions of Laguerre polynomials from the lie group view point, Bull. Cal. Math. Soc., 82, pp. 527-532, (1990).
[19] Alam, S. and Chongdar, A. K., On generating functions of modified Laguerre polynomials, Rev. Real Academia de Ciencias Zaragoza, 62, pp. 91-98, (2007).
[20] Das, S. and Chatterjea, S. K., On a partial differential operator for Laguerre polynomials, Pure Math. Manuscript,4, pp. 187-193, (1985).
[21] Chongdar, A. K., Some generating functions involving Laguerre polynomials, Bull. Cal. Math. Soc., 76, pp. 262-269, (1984).
[22] Patil, K. R. and Thakare, N. K., Multilinear generating function for the Konhauser biorthogonal polynomial sets,SIAM J. Math. Anal., 9, pp. 921-923, (1978).
[2] Konhauser, J. D. E., Biorthogonal polynomials suggested by the Laguerre polynomials, Pacific J. Math., 21, pp. 303-314, (1967).
[3] Spencer, L. and Fano, U., Penetration and diffusion of X-rays , calculation of spatial distribution by polynomial expansion, J. Res. Nat. Bur. Standards 46, pp. 446-461, (1951).
[4] Erdelyi, Arthur, with Magnus, W., Oberhettinger, F., Tricomi, F. G., et al., Higher transcendental function, vol. 2. New York: McGraw-Hill, (1953).
[5] Carlitz, L., A note on certain biorthogonal polynomials, Pacific J. Math., 24, pp. 425-430 (1968).
[6] Carlitz, L., A note on the Laguerre polynomials, Michigan Math. J., 7(3), pp. 219-223, (1960).
[7] Al-Salam, W. A., Operational representations for the Laguerre and other polynomials, Duke Math. Jour., 31, pp. 127-142, (1964).
[8] Calvez, L. C. et Genin, R., Applications des relations entre les fonctions generatrices et les formules de type Rodrigues, C. R. Acad. Sci. Paris Ser. A-B, 270, pp. A41-A44, (1970).
[9] Andrews,L.C., Special for Engineers and Applied Mathematicians, Macmillan Publishing Company.
[10] Prabhakar, T. R., On a set of polynomials suggested by the Laguerre polynomials, Pacific J. Math., 35 , pp. 213-219, (1970).
[11] Prabhakar, T. R., On the other set of biorthogonal polynomials suggested by Laguerre polynomials, Pacific J. Math., 37, pp. 801-804, (1971).
[12] Srivastava, H. M., Some Biorthogonal polynomials suggested by the Laguerre polynomials, Pacific J. Math., 98, pp. 235-250, (1982).
[13] Srivastava, H. M., A note on the Konhauser sets of Biorthogonal polynomials suggested by the Laguerre polynomials, Pacific J. Math., 91, pp. 235-250, (1980).
[14] Srivastava, A. N. and Singh, S. N., On the Konhauser polynomials Yna (x; k), Indian J. pure appl. Math., 10, pp. 1121-1126, (1979).
[15] Shreshtha, R.M. and Bajracharya, S., Group theoretical study of a bilateral generating function, Int. Trans. and Spec. func., 5(1-2), pp. 147-152, (1997), 147-152.
[16] Truesdell, C., An essay toward a unified theory of special functions, Princeton university press, (1948).
[17] McBride, E. B., Obtaining Generating Functions, Springer Verlag, Berlin, (1972).
[18] Sharma, R. and Chongdar, A. K., Some generating functions of Laguerre polynomials from the lie group view point, Bull. Cal. Math. Soc., 82, pp. 527-532, (1990).
[19] Alam, S. and Chongdar, A. K., On generating functions of modified Laguerre polynomials, Rev. Real Academia de Ciencias Zaragoza, 62, pp. 91-98, (2007).
[20] Das, S. and Chatterjea, S. K., On a partial differential operator for Laguerre polynomials, Pure Math. Manuscript,4, pp. 187-193, (1985).
[21] Chongdar, A. K., Some generating functions involving Laguerre polynomials, Bull. Cal. Math. Soc., 76, pp. 262-269, (1984).
[22] Patil, K. R. and Thakare, N. K., Multilinear generating function for the Konhauser biorthogonal polynomial sets,SIAM J. Math. Anal., 9, pp. 921-923, (1978).
Published
2017-03-23
How to Cite
[1]
K. P. Samanta and A. K. Chongdar, “On generating functions of biorthogonal polynomials suggested by the Laguerre polynomials”, Proyecciones (Antofagasta, On line), vol. 33, no. 2, pp. 133-146, Mar. 2017.
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