Scholarly Research Journal for Interdisciplinary Studies, Online ISSN 2278-8808, SJIF 2016 = 6.17, www.srjis.com UGC Approved Sr. No.49366, NOV-DEC 2017, VOL- 4/37 https://doi.org/10.21922/srjis.v4i37.10531
A STUDY ON GENERALIZED HERMITE POLYNOMIALS Kamal Gupta P.G. Department of Mathematics, GuruNanak College, Ferozepur Cantt.
In this paper, we obtain generating functions involving hyper geometric functions. Rodrigues type formula of Hermite polynomials which is closely related to generalized Hermite polynomials of Dattoli et. al. These results provide useful extensions of the well known results of classical Hermite polynomials Hn(x). Keywords: Generating functions, Hermite polynomials, Gould-Hopper polynomials, Rodrigues type formula,Dattoli et.al. AMS(MOS): Subject Classification (2000): 33 Special Functions, 33C45, 33C80 Scholarly Research Journal's is licensed Based on a work at www.srjis.com
Introduction: The Gould-Hopper polynomials g m n ( x, y) [2; p.512 (18)] see also [3; p.58 (6.2)] are generalization of classical Hermite polynomials Hn(x) [6; p.187(2)]. The notation
H (nm) (x, y) for g (nm ) (x, y) was givenby Dattoli et.al. and by Pathan, Yasmeen and Qureshi [4]. It is defined by [3; 6; p.76(6)] ( m) gm n ( x , y) H n ( x , y)
[ n / m]
k 0
n! y k x n mk k! (n mk )!
…(1.1) m (m;n; m x m F0 y . ; x n
… (1.2)
where m is a positive integer and (m; -n) abbreviates the array of m parameters,
n m 1 n n 1 , , ..., ; m 1. m m m These polynomials reduce, when m = 2 and y = 1, to the classical Hermite polynomials. The equation (1.1) can be derived from the following generating relations [2; p.512 (19)]
H (nm) (x, y)
n 0
tn exp ( xt ytm ) n!
… (1.3)
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Kamal Gupta 8289 (Pg. 8288-8292) and reduces to Hermite polynomials of two variables [2; p.511(11)]
k 0
(1) k n!(2x ) n 2k y k … (1.4) k!(n 2k )!
H (n2) (2x, y)
… (1.5)
H n ( x , y)
[ n / 2]
which can be derived from the generating relations [2; p.510(8)]
H n ( x , y) t n exp (2xt yt2 ) n! n 0
… (1.6)
H n (x,1) H n (x)
… (1.7)
In this paper we shall give some basic relations and properties involving the generalized Hermite polynomials Hn(x,y) and then take up generating function and Rodrigues type formula for Hn (x, y) and Hn (x) are derived as special cases. Generating Functions Theorem – 1 Any values of parameters and variables leading to result which do not make sense are tactily excluded then
(c) n H n (x, y)
n 0
tn (1 2xt ) c n!
c c 1 2 2 , 2 ; 4 yt ; 2xt 1 F 2 0 (1 2xt ) 2 ;
…(2.1)
Proof: Consider the following series
( c) n H n ( x , y )
n 0
[ n / 2]
n 0
k 0
tn n!
(c) n (1) k (2x ) n 2 k y k t n k! (n 2k )!
Replacing n by n+2k, and using Legendre’s duplication formula, we get
(1 2xt ) c2k
k 0
(1 2xt )
c
c c 1 2 k (4 yt ) 2 k 2 k k!
c c 1 2 , ; 4 yt 2 F0 2 2 2 ; (1 2xt )
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Kamal Gupta 8290 (Pg. 8288-8292) which is required result (2.1). Theorem-2 Any values of parameters and variables leading to result which do not make sense are tactily excluded, then
tn exp (2xt yt2 ) n!
2 F0 [n, c;; v]H n
( x , y)
x1 2( x yt) vt
c c 1 4y v 2 t 2 Proof: Consider the following series , ;; 2 2 2 2 ( 1 2 xvt 2 yvt )
n 0
c
2 F0
(c) k H k ( x yt, y)( vt) k S k! k 0
…(2.3)
Now using [4; p.452 (2.4)], we get S exp (2xt yt2 )
n 0
(1) k (c) k v k H n k ( x, y) t n k . k! n! k 0 ]
Replacing n by n-k and using
(n ) k
(1) k n! ,0 k n, we get (n k )!
tn . n!
… (2.4)
c c 1 4y v 2 t 2 S [1 2v( x yt) t ]c 2 F0 , ;; 2 2 (1 2xvt 2 yvt ) 2 2
… (2.5)
S exp (2xt yt2 ) 2 F0 [n, c; ; ] H n ( x, y) n 0
Now again by (2.3) and using (2.1), we get
Now equating the equation (2.4) and (2.5), we get required result (2.2). Rodrigues type formula Theorem-3 If D
y , then Hn (x,y) = 2n exp D 2 x n x 4
… (2.6) Proof: Again by generating relations (1.6), we get
tn ( yt2 ) n H n (x, y) n! exp (2xt ) n! n 0 n 0
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Kamal Gupta 8291 (Pg. 8288-8292) n
1 y 2 D exp (2xt ) n 0 n! 4 n y 2 (2xt ) exp D 4 n 0 n!
Equating the coefficient of tn, we get required result (2.6). Theorem-4 Use the fact that exp (2xt – yt2) = exp [2(xt) – y(xt)2] exp (yt2 (x2-1)) … (2.7) To prove H n ( x , y)
[ n / 2]
s 0
n!H n 2s (1, y) x n 2s y s ( x 2 1) s s! (n 2s)!
… (2.8)
Proof: By (2.7) we can write
H n ( x , y)
n 0
tn n!
( xt ) n n!
H n (l, y) n 0
y s t 2s ( x 2 1) s s! s 0
Replacing n by n – 2s, in right hand side and equating the coefficient of tn, we get required result. Special Cases I.
For y = 1, then (2.1) reduced to [6; p.190 (1)], which is published by Brafman [1] and
for c = 1 was given by Truesdell [8]. For y = 1, v = y, then (2.2) reduces to
2 F0 [n, c;; y]
n 0
1 2 y ( x t )
c
H n (x) t n exp (2xt t 2 ) n! c c 1 4y 2 t 2 , ;; 2 2 (1 2xyt 2 yt ) 2 2
2 F0
This is a well known result [6; p.198] and obtained by Brafman [1] with contour integration as the main tool. II.
For y = 1, (2.6) reduce to [5;p.129(2)].
III.
For y = 1, then (2.8) reduces to
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Kamal Gupta 8292 (Pg. 8288-8292) H n (x)
[ n / 2]
s 0
n! H n 2s (1) x n 2s ( x 2 1) s s! (n 2s)!
This is a known result [7; p.133(3)] Special cases I to III are known formulae of generating functions, Rodrigues type formula for classical Hermite polynomial Hn(x). REFERENCES Brafman,F.; Some generating functions of Laguerre and Hermite polynomials. Canad.J.Math. 9, (1957), 180-187. Dattoli,G., Chiccoli,C., Lorenzutta, S.Torre, A., and Maino, G.; Generalized Bessel Functions and Generalized Hermite polynomials, Jour. Math. Anal. Appl., 178 (1993), 509-516. Gould, H.W. and Hopper, A.T.; Operational formulas connected with two generalization of Hermite polynomials. Duke Math. J.29, (1962), 51-63. Pathan, M.A., Yasmeen and Qureshi, M.I.; Linear and bilinear generating function involving GouldHopper polynomials, Math. Sce. Res. J. 6 (9) (2002), 449-456. Pathan, M.A., Kazim, M.A. and Saksena, K.M.; Elements of Special Functions, P.C. Dwadash Shreni and Co. Pvt. Ltd., Aligarh (1972). Rainville, E.D.; Special Functions. Macmillan, New York; Reprinted by Chelsea Pub. Co., Bronx, New York, 1971. Saxena, R.K. and Gokhroo, D.C.; Special Functions, J.P.H., Jaipur (1987). Truesdell, C.; A Unified Theory of Special Function. Princeton Univ. Press, Princeton, New Jersey (1948).
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