International Journal of Contemporary Mathematical Sciences Vol. 9, 2014, no. 11, 545 - 551 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijcms.2014.4781 Extensions of Legendre Polynomials Ghulam Farid Global Institute Lahore New Garden Town, Lahore, Pakistan G. M. Habibullah Global Institute Lahore New Garden Town, Lahore, Pakistan Copyright © 2014 Ghulam Farid and G. M. Habibullah. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract (,;)ar In this paper, we introduce a simple set Fxn () and as its particular 1 (,;)ac cases, a two parametric extension Fx2 ()and a three parametric extension n Fx(,;)a c r () of the classical Legendre polynomial Px(). We establish different n n forms of the extended polynomials, generating functions, basic recurrence relations, the pure recurrence relations, the second order differential equations, Rodrigues formulae and orthogonality. Mathematics Subject Classification: 33C45, 11B37, 05A15 Keywords: Legendre polynomial, extended Legendre polynomial, recurrence relation, generating function, Rodrigues formula, orthogonality 1 Introduction Legendre polynomials are one of the most significant classical orthogonal polynomials arisen from the expressions of the form 546 Ghulam Farid and G. M. Habibullah 1 2 2 n (1 2xt t ) Pn ( x ) t . n0 Large dedicated literature is available to study orthogonal polynomials. We refer some of them beginning with [1], followed by [2], [11], [6], [8] and [3]. We also refer some recent work regarding properties, extensions, generalizations and applications of Legendre and the related orthogonal polynomials; e.g. [7], [5], [10] and [9]. For any two non-zero real numbers a and r , and a real valued continuous (,;)ar function (x ) 0 , following [4], we define Fxn ()as 2r ( a , ; r ) n (1) (1axt ( x ) t ) Fn ( x ) t . (1.1) n0 1 (,;)a 1 For our convenience, we denote Fx2 (), the trivial case when r , n 2 (,)a by Fxn (). It follows from (1.1) that n ( (xr ))k ( ) F(,;)a r()(). x t n n ax n k t n k n k!( n k )! n0 n 0 k 0 Lemma 10 and Lemma 11, pp. 56-58 of [1] will yield n 2 k (a , ; r ) n( (xr )) ( )nk n 2 k n Fn ()(). x t ax t n0 n 0 k 0 k!( n 2 k )! Hence, it follows by comparison that n 2 k (a , ; r )( (xr )) ( )nk n 2 k (2) Fn ()(). x ax (1.2) k0 k!( n 2 k )! p We observe from (1.2) that for each p 0,1,2 , Fx(,;)a cx r () is a n polynomial of degree precisely n in x if r is neither zero nor a negative integer. p This means that Fx(,;)a cx r () is a simple set for each (,)p r A B , where n n A{0,1,2}, B x: ( x )nk 0, 0 k . 2 Also if ()xc , then relation (1.2) shows that ()r (3) F(,;)a c r()() xn ax n , (1.3) nnn! 2 such that (r ) 0 and is a polynomial of degree n 2. Moreover, n n2 (4) (,;)(,;)a c r n a c r (5) Fnn( x ) ( 1) F ( x ) . (1.4) Extensions of Legendre polynomials 547 2 Main Results Since the proofs of are traditional oriented, we list the properties relating the extended polynomials. The properties involve different forms, generating functions, basic recurrence relations, the pure recurrence relations, the second order differential equations, Rodrigues formulae and orthogonality for the extended Legendre polynomials. n 2 k 1 2 ()()cx nk (a , cx2 )2 n 2 k (6) Fn ()(), x ax (2.1) k0 k!( n 2 k )! (,)(,)a cx22 a cx (7) x Fnn( x ) n F ( x ) 0, (2.2) n k1 n2 k 1 ()()ca 2 nk 22 2 nn 2 (8) (1axt cx t ) Cnn x t , C , (2.3) nk00k!( n 2 k )! n k 1 2 ()()c nk (a , c )2 n 2 k (9) Fn ()(), x ax (2.4) k0 k!( n 2 k )! 1 dan 2 (10) F(a , c )()() x x 2 c n , (2.5) n n!4 ann dx k n k n n (ac , ) 2 ax22 c ax c Fn ( x ) ( C ) , (2.6) k k0 44 (,)(,)(,)a c a c a c (11) ax Fn( x ) 2 c F n1 ( x ) an F n ( x ), (2.7) (,)(,)(,)a c a c a c (12) a(21) n Fn ()2 x F n11 () x 2 c F n (), x (2.8) 22 (,)a c 2 (,) a c (,) a c (13) (a x 4) c F () x a nx F ()2 x nac F (), x (2.9) n n n1 (14) 2nF(,)(,)(,)a c () xan (2 1) xF a c ()2( x cn 1) F a c (), x (2.10) n n12 n 24c (a , c ) ( a , c ) ( a , c ) (15) (x2 ) Fn ()2 x x F n () x n (1) n F n ()0. x (2.11) a Also, for each fixed (,)ac 2 , (ac , ) (0,0), c 0 ; the extended (,)ac Legendre polynomials set Fxn () is an orthogonal set satisfying the relations c 2 a (16) F(,)(,)a c( x ) F a c ( x ) dx 0, n m, (2.12) nm c 2 a 548 Ghulam Farid and G. M. Habibullah c 1 2 m a 2 4 c 2 (17) F(,)ac (), x dx (2.13) m c am(2 1) 2 a c 1 2 m a 2 m(,) a c 2!cm (18) x Fm ( x ) dx m . (2.14) m1 1 c 2 a ()m1 a 2 Moreover, Fx(,)ac () ax22 c ax c (19) n tn F ;1; t F ;1; t c , (2.15) 2 0 1 0 1 n0 (n !) 4 4 2 22a t() x c (a , c ) n a 1 1 4 (20) Fn ( x ) t (1 xt ) 10 F ; ; , (2.16) 22a 2 n0 (1 xt ) 2 n 2 2 2k n 2 k (,)ac n! ( a x 4 c ) ( ax ) (21) Fxn (), nk 22 (2.17) 2k0 (n 2 k )!( k !) 2 tn b b1 t2 ( a 2 x 2 4 c ) (22) n(,) a c b (2.18) 2 (b )nn F ( x ) (1 axt )21 F , ;1;2 , b , n0 n! 2 2 (1 axt ) n 2 2 2 n(,) a ct axt t( a x 4 c ) (23) 2Fn ( x ) e01 F ;1; , (2.19) n0 n!4 n n( a , c )t axt 2 2 (24) 2Fn ( x ) e J0 ( t 4 c a x ), (2.20) n0 n! n (,)ac 2 2 c ax (25) Fn ( x ) c21 F ( n , n 1;1; ), c 0. (2.21) 4 (,;)a c r The three parametric extended Legendre polynomial Fxn ()is given by n 2 k (a , c ; r )()()crnk n 2 k (26) Fn ()() x ax , (2.22) k0 k!( n 2 k )! arn () (27) F(,;)a c r() xn x n , (2.23) nnn! 2 where is a polynomial of degree n 2 and n2 (28) F(,;)(,;)a c r( x ) ( 1) n F a c r ( x ). (2.24) nn The extended Legendre polynomial Fx(,;)a c r () has the following n properties; 12 k for each n , if r1 k , r , then 2 Extensions of Legendre polynomials 549 aa2 n ()()xn22 k x c k 2 1 (29) F(,;)a c r ( x ) (2 r ) 24 , (2.25) nn 1 k0 22k (r ) (n 2 k )! k ! 2 k 1 n and for each , if r then 2 (,;)a c r a 22 xt Fxn () n 2 14a x c 2 (30) t e01 F ;; r t , (2.26) n0 (2r )n 2 16 ()() Fx(,;)a c r a1 1 a22 x 4 c (31) nn n 2 , (2.27) t(1 xt )21 F , ; r ; 2 t n0 (2r )n 2 2 2 2 (2 axt ) n 11 (2r ) (,)rr a (32) F(,;)a c r ()() x c2n P 2 2 x , (2.28) nn1 ()r 2 c 2 n or, equivalently 1 (,;)a c r (,)rr (r 1)n 2 2 c (33) Pnn()() x n F x . (2.29) 2 a cr(2 1)n Also, 2 1 r n a 2 2 ( 1) (1x ) n 2 1 c(2r ) d a nr (34) F(a , c ; r )( x ) ( ) n n 4c (1 x 2 ) 2 , (2.30) n 1 n 2()r n ! dx 4 c 2 n and (,;)(,;)(,;)a c r a c r a c r (35) ax Fn( x ) 2 c F n1 ( x ) an F n ( x ), (2.31) (,;)(,;)(,;)a c r a c r a c r (36) a()()()(), n r F x F x c F x (2.32) n n11 n (a22 x 4) c F (,;)a c r () x a 2 nx F (,;) a c r ()2(21) x ac n r F (,;) a c r (), x (2.33) n n n1 2nF(,;)(,;)(,;)a c r ()2( x anrxF 1) a c r ()2(22) x cnr F a c r (). x (2.34) n n12 n 4c 2 (,;)a c r (,;) a c r (,;) a c r (2.35) (x2 ) Fn ()(12) x rxF n () xnnrF (2) n ()0. x a For each fixed (,,)a c r 3 , (a , c , r ) (0,0,0), c 0 ; the extended Legendre (,;)a c r polynomials set Fxn () is orthogonal satisfying the relation c 2 a 1 4c r (37) (x2 )2 F (a , c ; r ) () x F ( a , c ; r ) () x dx 0, n m.