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

In this paper, we introduce a simple set Fx(,;)ar () and as its particular  n  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 2)(xt  ) tP . x t  n n0 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 function (x )0 , following [4], we define Fx(,;)ar ()as n  2( ,ra ; ) r n  (1) (1(axt ) )( ) x tF x tn . (1.1) n0 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 )( tax nn )k n t k . n  n  k!( n k )! n0 n  0 k  0 Lemma 10 and Lemma 11, pp. 56-58 of [1] will yield n 2 k (a , ; r )2 nn k n ( (xr )) ( )nk  Fn ( x )( tax )  t . n0 n  0 k  0 k!( n 2 k )! Hence, it follows by comparison that n 2 k (a , ; rn )2 k ( (xr )) ( )nk  (2) Fn ( xax )(  ) . (1.2) k0 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()() xn ax n  , (1.3) nnn! 2 such that (r ) 0 and  is a polynomial of degree n  2. Moreover, n n2 (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 2 nk (6) F(a , cxn )2( kxax )( ) , 2  (2.1) n  k!( n 2 k )! k0 22 (7) x F(a , cxa )( cx(x , )( )  ) n 0, F x (2.2)  nn

n k1 n2 k 1 ()()ca   2 nk 22 2 nn 2 (8) (1), axt ,  cx t  Cnn x t C (2.3) nk00k!( n 2 k )!

n k 1 2 ()()c nk (a , c )2 n 2 k (9) Fn ()(), x  ax (2.4) k0 k!( n 2 k )! 1 dan 2 (10) F(a , cn )2( xx )( c ) , (2.5) n n!4 ann dx k n k n n (ac , ) 2 ax22 c   ax c  Fn ( x )  ( C ),     (2.6) k 44    k0     (,)(,)(,)a c a c a c (11) ax F( x ) 2 c F ( x ) an F ( x ), (2.7)  n  n1  n (,)(,)(,)a c a c a c (12) a(21) n F ()2 x  F () x  2 c F (), x (2.8) n n11  n  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 n1 (14) 2nF(,)(,)(,)a c () xan (2  1) xF a c ()2( x  cn  1) F a c (), x (2.10) nn n 12 24c (a , c ) ( a , c )  ( a , c ) (15) (x2 ) 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 )  dxm  .  (2.14)  m1 1 c 2 a ()m1 a 2 Moreover,  Fx(,)ac () ax22 c   ax c  (19) n tn  F ;1; t F  ;1; t  c   , (2.15)  2 0 1  0 1   n0 (n !) 4   4  2 22a  t() x c (a , c )1 n a  1 4 (20)  Fn ( x ) t (1  xt ) 10 F  ; ;, (2.16) 22a 2 n0 (1 xt ) 2 n 2 2 2k n 2 k (,)ac n! ( a x 4 c ) ( ax ) (21) Fxn ( ), nk 22 (2.17) 2k0 (n 2 k )!( k !) 2  tn  b b1 t2 ( a 2 x 2 4 c ) (22) n(,) a c b (2.18) 2 (b )nn F ( x ) (1  axt )21 Fb , ;1;2 , , n0 naxt! 2 2 (1 )  n 2 2 2 n(,) a ct axt  t( a x 4 c ) (23) 2Fn ( x ) e01 F  ;1; , (2.19)  n!4 n0   n n( a , c )2 2 t axt (24) 2Fn ( x ) e J0 ( t 4 c a x ), (2.20) n0 n! n (,)ac 2 2 c ax (25) Fn ( x ) c21 F (  n , nc  1;1; ), 0. (2.21) 4 The three parametric extended Legendre polynomial Fx(,;)a c r ()is given by n n 2 k (a , c ; rn )2 k ()()crnk  (26) Fn ( xax )(  ) , (2.22) k0 k!( n 2 k )! arn () (27) F(,;)a c r() xn x n  , (2.23) nnn! 2 where  is a polynomial of degree n  2 and n2 (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 r1  k , r  , then 2

Extensions of Legendre polynomials 549

aa2 n ()()xn22 kk x c 2 1 (29) F(,;)a c r ( x ) (2 r ) 24 , (2.25) nn 1 k0 22k (r  ) (n 2 k )! k ! 2 k 1 n and for each n , if r  then 2 (,;)a c r a 22  xt Fxn () n 2 14a x c 2 (30)  t e 01 F;; rt , (2.26) n0 (2r )2n 16   ()() Fx(,;)a c r aa x c1 122 4 (31) nn n  2 , (2.27) t(1  xt )21 F , ; rt ; 2  (2raxt )2 2 2 2 (2 )  n0 n  n 11 (2r ) (,)rr a (32) F(,;)a c r ( x )( cPx22 ) 2 n , (2.28) nn1 ()r  2 c 2 n or, equivalently 1 (,;)a c r (,)rr (r  1)n 2 2 c (33) Pnn( xF )( n ) x . (2.29) 2 a cr(2 1)n Also, 2 1 r n a 2 2 ( 1) (1x ) n 2 1 cd(2r ) a nr (34) F(a , c ; r )2( xx ) ( )(1 n n ) 4c 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 F( x ) 2 c F ( x ) an F ( x ), (2.31)  n  n1  n (,;)(,;)(,;)a c r a c r a c r (36) a()()()(), n r F x  F x  c F x (2.32) n n11  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 nn 1 2nF(,;)(,;)(,;)a c r ()2( x anrxF   1) a c r ()2(22) x  cnr   F a c r (). x (2.34) nnn 12 4c 2 (,;)a c r (,;) a c r  (,;) a c r (2.35) (x2 ) 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 polynomials set Fx(,;)a c r () is orthogonal satisfying the relation  n  c 2 a 1 4c r (37) (x2 )2 F (a , c ; r ) () x F ( a , c ; r ) () x dx  0, n  m . (2.36)  2 nm c a 2 a

550 Ghulam Farid and G. M. Habibullah

Just to illustrate the arguments used there in, we prove results (2.5) and (2.12). Theorem 2.1: (Rodrigues formula) Rodrigues formula of the extended polynomial Fx(,)ac ()is given by n 1 dan 2 F(a , cn )2( xx )() . c n n!4 ann dx Proof: It follows directly from (2.4) that n 2 k (a , cn )2 k (cn ) k (2 2 )!  Fn ( xax )(  ) 22nk k0 k!( n 2 k )! ( n k )!2 1 dan n n 2 C()() ckn k x2  nn k n!4 a dx k0 n 2 1 da2 n (38) nn() . xc (2.37) n!4 a dx (,)ac Theorem 2.2: (Orthogonality of Fxn ()) For each fixed (,)ac 2 , (ac , )(0,0) , c  0 ; the extended Legendre polynomials set Fx(,)ac () is an orthogonal set satisfying the relation  n  c 2 a F(,)(,)a c( x ) F a c ( x ) dx 0, n m .  nm c 2 a Proof: By product rule of differentiation, it follows from (2.11) that  24c (a , ca )( c , )  (39) (x2 ) F nn () x  n (1) n F ()0. x (2.38) a  Similarly, we can write  24c (a , ca )( c , )  (40) (x2 ) Fmm  () x  m (1) m F ()0. x (2.39) a  Multiplying (2.38) by Fx(,)ac ()and (2.39) by Fx(,)ac ()and then subtracting the m n resulting relations, we have (,)(,)a c a c (n m )( n  m  1) Fnm ( x ) F ( x )  24c  (,)a c (,) a c (,) a c (,) a c ()()()()().x 2  Fm x F n x  F n x F m x  a  By integrating it follows that c 2 a (41) F(,)(,)a c( x ) F a c ( x ) dx 0, n m . (2.40)  nm c 2 a

Extensions of Legendre polynomials 551

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Received: July 29, 2014