on certain mixed generating functions of jacobi polynomials

Palestine Journal of Mathematics

Vol. 6(2)(2017) , 585–590

© Palestine Polytechnic University-PPU 2017

ON CERTAIN MIXED GENERATING FUNCTIONS OF JACOBI POLYNOMIALS N.U. Khan and T. Usman

Communicated by Ayman Badawi MSC 2010 Classications: 33C20, 33C45, 33C65, 33C90. Keywords and phrases: Legendre polynomials, Gegenbauer polynomials, Tchebcheff polynomials, Generalized Laguerre or Sonine polynomials, Jacobi and Lagrange's polynomials, Lauricella function, Appel's and generalized Gauss function, Pochhammer symbol. Abstract. This paper deals with general expansions which gives as special cases involving Jacobi and Laguerre polynomials, Lauricella, Appell and generalized Gauss function. The results unify and extended Exton's generating function [2] and Feldheim's expansion [3]. Also of interest are mixed generating functions which are partly unilateral and partly bilateral.

1 Introduction Now consider a generating function [7, p. 106 (11)]. ∞ ∑

Pn(α−n,β−n) (x)

n=0

[

1 t = exp − (x + 1)t (−α − β )n 2 n



]



1 F1 



−β ;

 t

−α − β ;

where Pn(α,β ) is a Jacobi polynomial [1; p. 170 (16)] 

Pn(α,β ) (x) =

(1 + α)n  2 F1  n!

(1.1)



−n, 1 + α + β + n ;

1  − (1 − x) 2

1+α;

(1.2)

or equivalently, Pn(α,β ) (x) =

n ∑ k=0

(1 + α)n (1 + α + β )n+k k ! (n − k )! (1 + α)k (1 + α + β )n

(

x−1 x

)k

(1.3)

.

where p Fq denote generalized hypergeometric function of one variable with p numerator parameters and q denominator parameters dened by [7; p. 42 (1)].  p Fq

 

(α1 ), (α2 ), · · · (αp ); (β1 ),

(β2 ), · · ·

(βq );

 ∞

 ∑ (α1 )n · · · (αp )n xn x = (β1 )n · · · (βq )n n! n=0

= p Fq (α1 , · · ·αp ; β1 , · · ·βq ; x)

where (a)n is the Pochhammer symbol, dened by {

(a)n =

1 a(a + 1)(a + 2) · · · (a + n − 1)

if n = 0 if n = 1, 2, 3, · · ·

(1.4) }

(1.5)

the denominator parameters are neither zero nor a negative integers the numerator parameters may be zero and negative integers. We list below a number of important polynomials which can be expressed in terms of Jacobi polynomial for different values of α and β .

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N.U. Khan and T. Usman

(µ− 12 ,µ− 12 )

Pn

(z ) =

(µ + 21 )n µ Cn (z ), (2µ)n

(1.6)

where Cnµ (z ) is the Gegenbauer polynomial (see [6],[7]), (− 21 ,− 12 )

Pn

( 1 , 12 )

Pn 2

( 12 )n Tn (z ), n!

(1.7)

( 23 )n Un (z ), (n + 1)!

(1.8)

(z ) =

(z ) =

where Tn (z ) and Un (z ) are the Tchebicheff polynomial of rst and second kind, (see [6],[7]) and Pn(0,0) (z ) = Pn (z ),

(1.9)

where Pn (z ) is the Legendre polynomial (see [6],[7]). Jacobi polynomial is an important class of orthogonal polynomial which is a generalization of ultraspherical polynomials. This class contains many special functions commonly encountered in the applications, e.g. Legendre, Gegenbauer, Tchebcheff, Laguerre and Bessel polynomials. Pathan and Kamarujjama [5; p. 2 (2.3)] and Khan [4; p. 67 (2.3)] introduced a generating relation involving product of three Laguerre polynomials in the form [ ( ( wv ) ] wv )γ exp − u + v − x (1 + u)α (1 + v )β 1 − u u ∞ ∞ ∞ ∑ ∑ ∑ α−(m+r ) β−(n−r ) = um v n L(m+r) (x)L(n−r) (x)L(rγ−r) (x)(−w)r m=−∞ n=m∗

(1.10)

r =0

and

(

exp u + v −

wv ) u

=

 

0 F1 

∞ ∑



; 1 + α;

∞ ∑

  − xu 0 F1 

(α)

m n

u v

m=−∞ n=m∗



; 1 + β;

(β )





  − xv  0 F1 

−

1 + γ;

xwv   u

(γ )

r ∞ L ∑ (m+r) (x)L(n−r ) (x)Lr (x)(−w ) r =0



;

(1 + α)m+r (1 + β )n−r (1 + γ )r

(1.11)

where m∗ = max(0, −m), so that all factorials of negative integers have meaning and (α) Ln (x) is known as generalized Laguerre or Sonine polynomial, dened as [6; p. 200 (1)] L(nα) (x) =

(1 + α)n 1 F1 (−n; 1 + α; x) n!

Equation (1.10) and (1.11) is infact generalization of number of results due to Feldheim [3]. Our work is motivated by a number of results of Exton [2], Feldhein [3], Pathan and Kamarujjama [5] and Khan [4] on Lagurre polynomials. The object of the present paper is to derive a general expansion for the product of Jacobi polynomial using series rearrangement technique.

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MIXED GENERATING FUNCTIONS OF JACOBI POLYNOMIALS

2 Generating Relation for the Product of Jacobi Polynomials In this paper we derive a generating relation involving the product of three Jacobi polynomials which generalize many known result of Feldheim [3] and Exton [2]. To obtain our main results, consider the product [ ( ] wv ) 1 (1 + x) 1 F1 [−β1 ; −α1 − β1 ; u] S (u, v, w) = exp − u + v − u 2 [ wv ] ×1 F1 [−β2 ; −α2 − β2 ; v ] 1 F1 −β3 ; −α3 − β3 ; u

(2.1)

Now expanding the right hand member of (2.1) as a multiple series with the help of (1.1), we get S (u, v, w) = ∞ ∑ ∞ ∑ ∞ (α −s,β1 −s) (α −k,β2 −k) (α −r,β3 −r ) ∑ Ps 1 (x) P 2 (x) Pr 3 (x) k

(−α1 − β1 )s (−α2 − β2 )k (−α3 − β3 )r

s=0 k=0 r =0

us−r v k+r (−w)r

(2.2)

Replacing s-r and k+r by m and n respectively then after rearrangement justied by the absolute convergence of the above series, it follows that

S (u, v, w) =

∞ ∑ ∞ ∑ m=0

×

um v n

n=m∗

∞ (α −(m+r ),β1 −(m+r)) (α2 −(n−r),β2 −(n−r)) (α −r,β3 −r ) ∑ Pm+1 r (x) Pn−r (x) Pr 3 (x)(−w)r

(−α1 − β1 )m+r (−α2 − β2 )n−r (−α3 − β3 )r

r =0

(2.3)

where m∗ = max(0, −m), so that all factorial of negative integers have meaning.

3 Special Cases Equation (2.3) gives many generating functions for well known polynomials. We are presenting only some interesting special cases here. (i)

Setting x = 1 in (2.3) and using the result [6, p. 254 (1)], we get Pn(α,β ) (1) =

(1 + α)n n!

(3.1)

[ [ ( wv ] wv )] exp − u + v − 1 F1 [−β1 ; −α1 − β1 ; u]1 F1 [−β2 ; −α2 − β2 ; v ] 1 F1 −β3 ; −α3 − β3 ; u u ∞ ∑

∞ ∑ um v n G(1 + α1 )G(1 + α2 ) = m ! n ! G ( 1 + α1 − m)G(1 + α2 − n) m=−∞ n=m∗



 ×4 F4 

−n, m − α1 , 1 + α2 + β2 − n, − α3 ;

1 + m, m − α1 − β1 , 1 + α2 − n, α3 − β3 ;

  w

(3.2)

588 (ii)

N.U. Khan and T. Usman

Setting x = −1 in (2.3) and using the result [6, p. 257], we get (−1)n (1 + β )n n! [ wv ] 1 F1 [−β1 ; −α1 − β1 ; u]1 F1 [−β2 ; −α2 − β2 ; v ] 1 F1 −β3 ; −α3 − β3 ; u Pn(α,β ) (−1) =

= (−1)

m+n

(iii)

(3.3)

∞ ∑

∞ ∑ um v n G(1 + β1 )G(1 + β2 ) m ! n ! G ( 1 + β − m ) G ( 1 + β2 − n)(−α1 − β1 )m (−α2 − β2 )n 1 m=−∞ n=m∗   −n, m − β1 , 1 + α2 + β2 − n, − β3 ;   ×4 F 4  w (3.4) 1 + m, m − α1 − β1 , 1 + β2 − n, α3 − β3 ;

In view of the relation [7; p. 441 (16)]

) ( x+1 x−1 ,− Pn(α−n,β−n) (x) = gn−α,−β −

2

2

(3.5)

equation (2.3) yields an interesting result [ ( ] wv ) 1 exp − u + v − (1 + x) 1 F1 [−β1 ; −α1 − β1 ; u]1 F1 [−β2 ; −α2 − β2 ; v ] u 2 ) (−α ,−β ) ( ∞ ∞ ∞ 1 x−1 [ ∑ ∑ ∑ gm+r1 1 − x+ wv ] m n 2 ,− 2 = u v 1 F1 −β3 ; −α3 − β3 ; u (−α1 − β1 )m+r m=−∞ n=m∗ r =0

) (−α3 ,−β3 ) ( x+1 x−1 ) (−α ,−β ) ( 1 x−1 gn−r2 2 − x+ − 2 ,− 2 gr 2 ,− 2 (−α2 − β2 )n−r (−α3 − β3 )r

(3.6)

where gn(α,β ) is Lagrange's polynomial [1; p. 267 see also 7; p. 85 (25)]. On replacing u, v and w by ut, vt and wt respectively, multiply both side by tγ−1 in equation (2.3) and then taking Laplace transform, we get

(iv)

[ u v wv ] γ, −β1 , −β2 , −β3 ; −α1 − β1 , −α2 − β2 , −α3 − β3 ; , , z z z ∞ ∑ ∞ ∞ r ∑ ∑ (γ + m + n)r (−w) um v n (γ )m+n = ( −α − β ) 1 1 m+r (−α2 − β2 )n−r (1 + α3 )r n=m∗ (3)

z −γ FA

m=0

r =0

(α −(m+r ),β1 −(m+r)) ×Pm+1 r (x)

where

[( z=

(α −(n−r ),β2 −(n−r))

2 Pn−r

wv ) u+v− u

(

)

(x) Pr(α3 −r,β3 −r) (x)

(3.7)

]

1+x −1 2

and FA(n) is Lauricella function dened by [7; p. 60 (1)] FA [a, b1 , b2 , · · · , bn ; c1 , c2 , · · · , cn ; x1 , · · · , xn ] (n)

=

∞ ∑ m1 ,...,mn =0

1 (a)m1 +···+mn (b1 )m1 · · · (bn )mn xm xmn 1 ··· n , (c1 )m1 · · · (cn )mn m1 ! mn !

(3.8)

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MIXED GENERATING FUNCTIONS OF JACOBI POLYNOMIALS

| x1 | + · · · + | xn |< 1 .

(v)

If we set w = 0, equation (2.3) becomes [

]

1 exp − (u + v ) (1 + x) 2 =

∞ ∑ ∞ ∑ m=0 n=m∗

(vi)

1 F1 [−β1 ; −α1

− β1 ; u]1 F1 [−β2 ; −α2 − β2 ; v ]

um v n P (α1 −m,β1 −m) (x) Pn(α2 −n,β2 −n) (x) (−α1 − β1 )m (−α2 − β2 )n m

(3.9)

For x = 0, equation (3.9) reduces to [ ] 1 exp − (u + v ) 1 F1 [−β1 ; −α1 − β1 ; u]1 F1 [−β2 ; −α2 − β2 ; v ]

2

 −n, − α1 ; ∞ ∑ ∞ m n ∑ u v  = 2 F1  m! n! m=0 n=m∗ − α1 − β1 ;





1   F  2 2 1



−n, − α2 ;

1  2

− α2 − β2 ;

(3.10)

On replacing u and v by ut and vt, multiplying both side by tγ−1 , and then taking Laplace transform, in equation (3.2), (3.4) and (3.10), we get

(vii)

(3)

FA

=

[ u v wv ] γ, −β1 , −β2 , −β3 ; −α1 − β1 , −α2 − β2 , −α3 − β3 ; , , z z z

∞ ∑ ∞ ∑ um v n G(1 + α1 )G(1 + α2 ) (γ )m+n m ! n! (−α1 − β1 )m (−α2 − β2 )n G(1 + α1 − m)G(1 + α2 − n) n=m∗

m=0

  ×5 F4 

−n, m − α1 , 1 + α2 + β2 − n, − α3 , γ + m + n;

1 + m, m − α1 − β1 , 1 + α2 − n, α3 − β3 ;

  w

(3.11)

and (3)

FA

=

∞ ∑ ∞ ∑

[ u v wv ] γ, −β1 , −β2 , −β3 ; −α1 − β1 , −α2 − β2 , −α3 − β3 ; , , z z z

(−1)m+n

m=0 n=m∗

  ×5 F4 

um v n G(1 + β1 )G(1 + β2 ) (γ )m+n m! n! (−α1 − β1 )m (−α2 − β2 )n G(1 + β1 − m)G(1 + β2 − n)

−n, m − β1 , 1 + α2 + β2 − n, − β3 , γ + m + n;

1 + m, m − α1 − β1 , 1 + β2 − n, α3 − β3 ;

  w

(3.12)

590

N.U. Khan and T. Usman

and (u

2

+

v

2

)−γ −1 F2 [γ, −β1 , −β2 ; −α1 − β1 , −α2 − β2 ; u, v ] 

 −m, − α1 ; ∞ ∑ ∞ m n ∑ u v  = 2 F1  m! n! m=0 n=m∗ − α1 − β1 ;



1   F  2 2 1

−n, − α2 ; − α2 − β2 ;



1  2

(3.13)

where F2 is Appell function of two variables is dened by [7; p. 53 (5)] ∞ ∑

F2 [a, b, b′ ; c, c′ ; x, y ] =

m,n=0

(a)m+n (b)m (b′ )n xm y n (c)m (c′ )n m!n!

(3.14)

|x| + |y| < 1

References [1] A. Erde ´lyi, W. Magnus, F. Oberhettinger and F.G. Tricomi, Higher trancedental functions, vol. 3, McGraw Hill Book Co. Inc., New York (1955). [2] H. Exton, A new generating function for the associated Laguerre polynomials and resulting expansions, Jnanabha 13, 147–149 (1938). [3] E. Feldheim, Relation entre les polynomes de Jacobi, Laguerre et Hermite., Acta Math. 74, 117–138 (1941). [4] N.U. Khan, A new class of generating functions associated with the product of Laguerre polynomials, South East Asian J. Math. and Math. Sc. 3, No.2, 65–70 (2005). [5] M.A. Pathan and M. Kamarujjama, On certain mixed generating functions Journal of Analysis, Forum D'Analystes 6, 1–6 (1998). [6] E.D. Rainville, Special functions, The Macmillan Company, New York (1960). [7] H.M. Srivastava and H.L. Manocha, A Treatise on generating functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and sons, New York (1984).

Author information N.U. Khan and T. Usman, Department of Applied Mathematics, Faculty of Engineering and Technology, Aligarh Muslim University, Aligarh-202002, India. E-mail:

[email protected], [email protected]

Received: Accepted:

November 28, 2015. May 17, 2016.