Gegenbauer Polinomları İçin Bilineer ve Bilateral Doğurucu Fonksiyonlar

AKÜ FEMÜBİD 18 (2018) 011305 (162-168) AKU J. Sci. Eng. 18 (2018) 011305(162-168)

DOİ:

10.5578/fmbd.66800

Gegenbauer Polinomları İçin Bilineer ve Bilateral Doğurucu Fonksiyonlar

Nejla Özmen

Düzce Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Düzce.

e-posta: nejlaozmen06@gmail.com.

Geliş Tarihi:15.03.2017 ; Kabul Tarihi:10.04.2018

Anahtar kelimeler Doğurucu fonksiyon; Gegenbauer polinomları; Multilinear ve multilateral doğurucu fonksiyon. Özet

Bu çalışma Gegenbauer (veya ultraküresel) polinomları için bilineer ve bilateral doğurucu fonksiyonlarla ilgilidir. Gegenbauer polinomları için bazı özel durumları, çeşitli özelliklerini, multilinear ve multilateral doğurucu fonksiyonları elde edilmiştir.

Bilinear and Bilateral generating functions for the Gegenbauer

Polynomials

Keywords Generating function; Gegenbauer polynomials; Multilinear and multilateral generating function. Abstract

The present study deals with bilateral and bilinear generating functions for the Gegenbauer (or ultraspherical) polynomials. In this manuscript we obtain some special cases for Gegenbauer polynomials. miscellaneous properties and multilinear and multilateral generating functions.

© Afyon Kocatepe Üniversitesi

163

1. Introduction

Generalized functions oscupy the pride of place in literature on special functions. Their importance which is mounting everyday stems from the fact that they generalize well-know one variable special functions namely Hermite polynomials, Laguerre polynomials, Legendre polynomials, Gegenbauer polynomials, Jacobi polynomials, Rice polynomials, Generalized Sylvester polynomials etc. All these polynomials are closely associated with problems of applied nature. For example, Gegenbauer polynomials are deeply connected with axially symmetric potentials in dimensions and contain the Legendre and Chebyshev polynomials as special cases. The hypergeometric functions of which the Jacobi polynomials is a special case, is important in many cases of mathematics analysis and its applications.

The Gegenbauer (or ultraspherical) polynomials

C C :

 m

C can be defined by Gauss

hypergeometric series as follows (Olver et al. 2010) , 2 1 ; 2 , ! ) 2 ( : ) ( 2 1 1 2                    z m m F m z C_m m (1) for

m



N

0 and









1

/

2

,



  

\

0

.

The Gauss hypergeometric function

C N C C  D F : ( \ 0) 2 1 2 is defined as , ! ) ( ) ( ) ( : ; , 0 1 2 m z z F m m m m m















            

where the Pochammer symbol (rising factorial)

C

C





)

:

(

m is defined by ), 1 ( : ) ( 1   



 s m s m





where

m



N

0

.

Consider the generating function for Gegenbauer polynomials given by Olver et al. (2010), namely

. ) 2 1 ( ) ( 2 0        



Cm z tm zt t m (2) We attempt to generalize this expansion using the

representation of Gegenbauer polynomials in terms of Jacobi polynomials given by Olver et al. (2010), namely

.

)

(

)

(

)

2

(

)

(

( 1/2, 1/2) 2 1

P

z

z

C

_m m m m  





  





(3)

The Jacobi polynomials Pm(,) : CC can be

defined by Gauss hypergeometric series as follows (Srivastava and Monocha 1984):

,

;

1

;

1

,

!

)

1

(

:

)

(

1₂ 1 2 ) , (



































m z m

m

m

F

m

z

P









 

for

m



N

0 and



,



1 such that if



1

,

0



,









then







1 0.

Gegenbauer polynomials by suitably specializing the parameters in the corresponding results for the Jacobi polynomials: m m m m m

_C

₍

_z

₎

_t

)

(

)

(

0



  







,

,

;

;

(

2

1

)

,

(

2

1

)



,

1

z

z

t

z

z

t

F



















(4)

which, for



2



, yields

m m m m m

t

z

C

(

)

)

2

(

)

(

0 







 





, ; 2 ) 1 ( 1 1 2 ; , ) 1 ( 1 2 2 1 2 2                          t x x x t F t z z (5) or, alternatively,

164 m m m m m

t

z

C

(

)

)

2

(

)

(

0 







 









, ; 1 1 ; , ) 1 ( 2 1 2 2 2 2 1 2 1 2 1 1 2                        zt t z F zt (6)

view of the quadratic transformation (Srivastava and Monocha 1984):

 

. ; ; , ) 1 ( ; 2 2 ; , 2 1 2 1 2 1 2 1 2 1 1 2 1 2                           b a a F z b z b a F  z_z

Starting, as usual, from (2) we get the following formula of the type (1) for the polynomials

) (x

Cm



(Srivastava and Monocha 1984):

(

)

2

(

),

0





  

z

t

C

t

z

C

m

m

r

r r m r m m















 

    



(7) where



is defined by





(

1



2

zt 

t

2

)

1/2,

1





.

The Gegenbauer (or ultraspherical) polynomial

) (z C_m (



1/2,

z



1

)

are defined by Horadam (1985), , 2 ) ( , 1 ) ( ₁ 0 z C z z C   



with the recurrence relation

). 2 ( ) ( ) 2 2 ( ) ( ) 1 ( 2 ) ( 2 1        _{ } m z C m z C m z z mC m m m   





Lemma 1. The following addition formula holds for

the Gegenbauer (or ultraspherical) polynomials:

). ( ) ( ) ( 1 2 2 1 0 z C z C z C n m m n m n        _



(8) Proof. Replacing



by



1





2 in (2), we obtain n m m n n m n m m m n n n n n n

t

z

C

z

C

t

z

C

t

z

C

t

zt

t

zt

t

zt

t

z

C

)

(

)

(

)

(

)

(

)

2

1

(

)

2

1

(

)

2

1

(

)

(

2 1 2 1 2 1 2 1 2 1 0 0 0 0 2 2 2 0                         































From the coefficients of tn on the both sides of the last equality, one can get the desired result.

2. Multilinear and Multilateral Generating Functions

We aim here at presenting a family of bilinear and bilateral generating relations for the Gegenbauer (or ultraspherical) polynomials C_m(x) which are

generated by (2) without using grouptheoretic technique but, with the help of the similar method as given in (Özmen and Erkuş-Duman 2013, 2015, 2018).

Theorem 1. For a non-vanishing function

) ,..., (s₁ s_r   of r complex variables

s ,...,

₁

s

_r )

(rN and for



,



C , mN; let

) 0 ( ) ,..., ( : ) ; ,..., ( 1 0 1 ,      _ 



m m r m m m r a s s a s s



 



  and





 

.

)

,...,

(

)

(

:

;

,...,

;

1 / 0 1 , , m r m sm n m s n m r s n

s

s

z

C

a

s

s

z





       









Then, for sN , we have



1



. ) ; ,..., ( ) 2 1 ( ; ,..., ; 1 , 2 1 , , 0               



w s s w zw w w s s z r l s r s l l





     (9)

165

Proof. Let E denote the first member of the

assertion (9) of Theorem1. Then,

.

)

,...,

(

)

(

₁ ] / [ 0 0 sm l m r m sm l m s l m l

w

s

s

z

C

a

E

     









  



Replacing

l

by l sm,



1



)

;

,...,

(

)

2

1

(

)

,...,

(

)

(

)

,...,

(

)

(

1 , 2 1 0 0 1 0 0



















          









w

s

s

w

zw

s

s

a

w

z

C

w

s

s

z

C

a

E

r m r m m m l l l l m r m l m m l







        

which completes the proof.

Theorem 2. For a non-vanishing function

) ,..., (y₁ y_r   of r complex variables

y ,...,

₁

y

_r )

(rN and of complex order



, let

,

)

,...,

(

)

(

:

)

;

,...,

;

(

1 ] / [ 0 1 , 2 1 2 1 s r ps m s p m s r

y

y

x

C

a

y

y

x

s





        







   



where

a

_s



0

,



,





C

, m,pN and the notation



m /

p



means the greatest integer less than or equal m /p . Then, we get l r l p k l m k

y

y

x

C

x

C

a

_m _k

(

)

_k _pl

(

)

_{ l}

(

₁

,...,

)



] / [ 0 0 2 1   







 

).

;

,...,

;

(

₁ , 2 1 



  

x

y

y

r 





(10) Proof. Let Hdenote the right side of the assetion (10). Then, upon subsituting for the Gegenbauer polynomials

_C

1 2

(

_x

)

pk m

 

 from the (8) into the

left-hand side of (10), we get   l r l k pl k m l pl m k p m l y y x C x C a H  ( )  ( )   ( 1,..., )



0 / 0 2 1        





  l r l k pl k m pl m k l p m l y y x C x C a  ( )  ( ) _{ }( 1,..., )



0 / 0 2 1              





  l r l pl m l p m l

y

y

x

C

a

 

(

)

_{ }

(

₁

,...,

)



/ 0 2 1    







).

;

,...,

;

(

₁ , 2 1 



  

x

y

y

r 





Theorem 3. For a non-vanishing function

) ,..., (s₁ s_r   of r complex variables

s ,...,

₁

s

_r )

(rN and of complex order



, suppose that





m r pm qm r m m r q p

s

s

z

C

a

s

s

z





   

)

,...,

(

)

(

:

;

,...,

;

1 0 1 , ,    









where

a

m



0

and ) ; ,..., ( 1 , ,pq s sr t 



 

.

)

,...,

(

:

1 / 0 k r pk k q m k

t

s

s

qk

m

m

r

a

_ 

























Then, for p,qN; we obtain

m q p r m z s s t C _r m





_  ) ; ,..., ( ) ( ₁ 0 , , 



 











 





  q r

t

s

s

z

q p r

)

(

;

,...,

;

₁ , , 2











   (11)



 (12x2)1/2,  1



.

Proof. For the proof of Teorem 3, we find it to be

convenient to denote the right side of the assertion (11) by



. Then,   . ) ,..., ( ) ( 1 / 0 0 m k r pk k q m k r m m t s s qk m m r a z C 



              





Replacing

m

by m qk and then using (7), we may write that

166 qk m k r pk k qk r m k m t s s a z C m qk m r                    







  ) ,..., ( ) ( 1 0 0 k q r pk k m qk r m m k t s s a z C m qk m r ) )( ,..., ( ) ( 1 0 0





                         





k q r pk qk r qk r k k

t

s

s

z

C

a

)

)(

,...,

(

)

(

1 2 0









         











k q q r pk qk r k k r

t

s

s

z

C

a

)

)(

,...,

(

)

(

1 0 2











        





















 





  q r q p r

t

s

s

z

)

(

;

,...,

;

₁ , , 2











  



 (12x2)1/2,  1



which completes the proof.

3.Special Cases

We can give promote apposition of the on teorems. Set

)

,...,

(

)

,...,

(

₁ ( 1,,..., ) ₁ r m r m

s

s

h

s

s

r      







in Theorem 1, where the multivariable extension of the Lagrange-Hermite polynomials

)

,...,

(

₁ ) ,..., ( 1 r m

s

s

h

 r 

 (Altın and Erkuş 2006),

generated by





 

_

_







r



r n r m m k k r k s s s z z s s h z s k r / 1 2 / 1 2 1 1 1 ,..., 0 1 ,..., , min ; . ,..., ) 1 ( 1           





C



   (12)

We obtain, the following result which provides a class of bilateral generating functions for the multivariable extension of the Lagrange-Hermite

polynomials ( 1,..., )

(

₁

,...,

)

r k

s

s

h

 r   and the Gegenbauer polynomials Cl (x)  . Corollary 1. If



0

,

,



,

)

,...,

(

:

)

;

,...,

(

1 ) ,..., ( 0 1 , 1

C









 _ 











      m m r m m m r

a

s

s

h

a

s

s

r then, we have l pm m r m sm l m s l m l

w

w

s

s

h

z

C

a

 r



  

)

,...,

(

)

(

( ,..., ) ₁ ] / [ 0 0 1     





). ; ,..., ( ) 2 1 ( , 1 2



   r s s w zw     (13) Remark 1. Taking

a

m



1

,



0, 1and using the generating relation (12) for the multivariable polynomials ( 1,..., )

(

₁

,...,

)

r m

s

s

h

 r   in Corollary 1, we have sm l m r m sm l s l m l

w

s

s

h

x

C

r     





   



)

,...,

(

)

(

1 ) ,..., , ( ] / [ 0 0 2 1



(

1

)



.

)

2

1

(

1 2 j j j r j

s

w

xw





  

_















min , 2 1/2,..., 1/ , 1



1 1   s  s  sr  r w



If we set r1, y1 y and

 

y

C

y

_s s

(

)

3     







in Theorem 2, we have the following summation

expression for the Gegenbauer polynomials.

Corollary 2. If

)

;

,...,

;

(

₁ , 2 1 



  

x

y

y

r 



 

,

)

(

:

1 2 3 ] / [ 0 s ps m s p m s

y

C

x

C

a

_ __s









then, we have

 

l y C x C x C a_{l m k k pl l} p k l m k       ₃ ) ( ) ( 2 1 ] / [ 0 0   





 

167

).

;

,...,

;

(

₁ , 2 1 



  

x

y

y

r 





(14) provided that each member of (14) exists.

Remark 2. Taking 𝑎𝑙 = 1,  0, 𝑝 = 1,



1 𝑦 =

𝑥, 𝜔 = 1 in Corollary 2 and using (8) we have

 

x C x C x C_{m k k l l} k l m k 3 ) ( ) ( 2 1 0 0     





  ). ( 3 2 1 _x C_m   If we set s1,s₁  y and

)

(

)

(

y

₁

P

( ,_k)

y

k      







in Theorem 3, where the classical Jacobi polynomials

P

_n(,)

(

y

)

is generated by (Erdélyi et al. 1955),

 



(

1

2

)

,

1



)

1

(

)

1

(

2

2 / 1 2 ) , ( 0



















    



t

t

xt

t

t

t

x

P

_n n n









     

we get a family of the bilateral generating functions for the classical Jacobi polynomials and the Gegenbauer (or ultraspherical) polynomials

) (x C_m as follows: Corollary 3. If





 



 N C



 



 _{ } 







_     , , , , 0 , ) ( : ; ; ( ) 0 , , , q p r a y P z C a y z m k pm qm r m q p m and  

 

.

:

)

;

(

( ) / 0 , , , k k q m k

t

y

P

qk

m

m

r

a

t

y

_pk q p    



_

_



















 Then, we get



(

1

2

)

,

1



.

)

(

;

;

)

;

(

)

(

2 / 1 2 0 , , 2 , ,



















 





  



 























    

x

t

y

z

t

y

z

C

q m q p r m q p r m

Furthermore, for each suitable choice of the coefficients

a

_m

(

m



N

₀

),

if the multivariable functions _k(y₁,...,y_r),

r



N

, are expressed as an appropriate product of a lot of simpler functions, the assertions of Theorem 1, Teorem 2, Teorem 3 can be applied to yield many different families of the Gegenbauer polynomials.

4. References

Altn A. and Erkus, E. 2006. On a multivariable extension of the Lagrange-Hermite polynomials, Integral Transforms and Special Functions, 17(4), 239-244.

Erdélyi, A., Magnus, W., Oberhettinger, F. , Tricomi, F. G. 1955. Higher Transcendental Functions, vol. III, McGraw-Hill Book Company, New York.

Horadam, A. F. 1985. Gegenbauer polynomials revisited, The Fibonacci Quarterly, 23(4), 294-299.

Olver, F. W. J. , Lozier, D. W. , Boisvert, R. F. , and Clark, C. W. 2010. NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge.

Özmen, N. and Erkus-Duman, E. 2013. Some results for a family of multivariable polynomials, AIP Conference Proceeding, 1558, 1124-1127. Özmen, N. and Erkus-Duman, E. 2015. On the

Poisson-Charlier polynomials, Serdica Mathematical Journal, 41(4), 457-470.

Özmen, N. and Erkus-Duman, E. 2018. Some families of generating functions for the generalized Cesáro polynomials, Journal of

168 Computational Analysis and Applications, 25(4),

670-683.

Srivastava, H. M. and Manocha, H. L. 1984. A Treatise on Generating Functions, Halsted Press, John Wiley and Sons, New York.