Some new properties of the Meixner polynomials

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SAKARYA ÜNİVERSİTESİ FEN BİLİMLERİ ENSTİTÜSÜ DERGİSİ SAKARYA UNIVERSITY JOURNAL OF SCIENCE

e-ISSN: 2147-835X

Dergi sayfası: http://dergipark.gov.tr/saufenbilder

Geliş/Received

27-07-2017 Doi

Kabul/Accepted 10.16984/saufenbilder.331327 21-09-2017

Some new properties of the Meixner polynomials

Nejla Özmen1 ABSTRACT

The present study deals with some new properties for the Meixner polynomials. In this manuscript we obtain a number of families of bilinear and bilateral generating functions, general properties and also some special cases for these polynomials. In addition, we derive a theorem giving certain families of bilateral generating functions for the generalized Lauricella functions and the Meixner polynomials. Finally, we get several interesting results of this theorem.

Keywords: Meixner polynomials, generating function, bilinear and bilateral generating function,

recurrence relations, hypergeometric function.

Meixner polinomlarının bazı yeni özellikleri

ÖZ

Bu çalışma Meixner polinomlar için bazı yeni özellikler ele alınmıştır. Burada elde edilen sonuçlar Meixner polinomların bilineer ve bilateral doğurucu fonksiyonların çeşitli ailelerini, çeşitli özelliklerini ve bazı özel durumlarını içermektedir. Bunlara ek olarak genelleştirilmiş Lauricella fonksiyonları ve Meixner polinomları için bilateral doğurucu fonksiyon içeren teorem verildi. Son olarak, bu teoremin ilginç bazı sonuçları verildi.

Anahtar Kelimeler: Meixner polinomları, doğurucu fonksiyon, bilineer ve bilateral doğurucu fonksiyon,

rekürans bağıntıları, hipergeometrik fonksiyon.

1 _{Matematik Bölümü, Düzce Üniversitesi - [email protected]}

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1. INTRODUCTION

The Meixner polynomials M_n(z;,x) are defined by the generating function relation (see, for example, [1], p. 449, Problem 20 (ii)) . ) 1 ( ) 1 ( ! ) , ; ( 0 z z n n n x u u n u x z M       



_  (1.1) It is from (1.1) that (see, [2]):

k n k n n x k n z k z n x z M                  



  0 ! ) 1 ( ) , ; ( (1.2)

In addition, we have the following relationship between the Meixner polynomials M_n(z;,x) and the classical Jacobi polynomials P_n(,)(x) ([1], p.443, Problem5(i)): ). 1 2 ( ! ) , ; (  ( 1,   )  x P n x z M_n  _n  n z

The following for the Meixner polynomials generating function relationship holds true [1]:

! ) , ; ( 0 n u x z M n m n n    



). 1 , ; ( ) 1 ( ) 1 ( u u x z M x u u z m z _m         (1.3) First of all, some of the definitions and notations used in this paper are presented here as follows:

The four Appell functions of bivariate function, denoted by F₁, F₂, F₃ and F (see [1], [4], [10], ₄, [14], [20]) were generalized by Lauricella functions of n variables which are denoted by F_A(n), (n)

B F , ) (n C F and (n) D F (see, [6], p. 60) and . , , , (2) ₃ (2) ₄ (2) ₁ 2 ) 2 ( F F F F F F F F_A  _B  _C  _D 

A further generalization of the well-known Kampé de Fériet hypergeometric function in bivariate function is owing to Srivastava and Daoust ([3], [4], [10], [14], [20]) defined the generalized Lauricella function as follows (see [3], [4] and [9, p. 37 et seq.]; see also [7, p. 106 et seq.] and [8, p. 143]):                       ; ) ( : ) ) ( ( ; ) ( : ) ) ( ( : ) ( ,..., ) ( : ) ( : ) ( ,..., ) ( : ) ( ) ( ;...; ) ( : ) ( ;...; ) ( : ₁ ₁ 1 1 1 1 1 1       d b n c n a n B B A n D D C F               n z z n n d n n b ,..., 1 ; ) ( : ) ) ( ( ; ) ( : ) ) ( ( ...; ...;   ! ... ! ) ,..., ( 0 ,..., 1 1 1 1 1 mn n m n z m m z n m m m m _n     

where, for convenience,

) ( ... ) 1 ( 1 1 ) ( ... ) 1 ( 1 1 ) ( ) ( : ) ( ₁,..., n j n m j m j C j n j n m j m j A j c a n m m          



  ) ( ) ( 1 ) ( ) ( 1 ) 1 ( 1 ) 1 ( 1 ) 1 ( 1 ) 1 ( 1 ) ( ) ( ... ) ( ) ( ) ( ) ( ) 1 ( ) 1 ( n j n m n j D j n j n m n j B j j m j D j j m j B j d b d b n n    



     the coefficients ), ( ), ( ,..., 1 ; ,..., 1 ,..., 1 ; ,..., 1 ) ( ) ( ) ( n B s n A s s s         





)

(

),

(

,..., 1 ; ,..., 1 ,..., 1 ; ,..., 1 ) ( ) ( ) ( n D s n C s s s         





are real constants and

 

( ) ) (  

B

b abbreviates the array of ) ( B parameters bs( )(s1,...,B( );1,...,n)   with similar interpretations for other sets of parameters (see [4], [10], [14], [20]). Here, as usual, ()_

denotes the Pochhammer symbol and

! ) 1 ( _n n, (0)₀:1, (n₀) is defined by . ; if ), 1 )...( 1 ( } 0 { \ ; 0 if , 1 ) \ ( ) ( ) ( ) ( 0                   C N C Z C             _ n n

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Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 21 (6), 1454~1462, 2017 For a suitably bounded non-vanishing multiple

sequence





0 N ,..., 2 1, ,..., ) ₁ ( _  s m m s m m m of real or

complex parameters, let _n(u₁;u₂,...,u_s) of s (real or complex) variables u1;u2,...,u_s defined by





! ... ! ,..., ), ..., , ( )) (( )) (( ) ( : ) ,..., ; ( 1 1 2 1 2 1 1 1 1 1 0 ,..., 2 0 1 s s s s s n m u m u m m m m f d b n u u u s m m m m m s m m n m    



      (1.4)

where, for convenience,

. ) ( )) (( and ) ( )) (( 1 1 1 1 1 1 j m m j m m j D j j B j d d b b _



_ _



_    

The main target, is to study different properties of the Meixner polynomials. Miscellaneous properties and different families of bilinear and bilateral generating functions, and also some special cases for these polynomials are given. In addition, we derive a theorem giving certain families of bilateral generating functions for the generalized Lauricella functions and the Meixner polynomials. Nowadays, there are a lot of works related to Meixner polynomials and Lauricella functions theory and its applications (see [17], [18], [20]).

Lemma 1.1. The following addition formula holds for the Meixner polynomialsM_n(z;,x):

) , ; (z₁ z₂ ₁ ₂ x M_n    ). , ; ( ) , ; ( ₁ ₁ ₂ ₂ 0 x z M x z M p n p p n n p          



(1.5)

Proof Replacing z by z₁z₂ and  by ₁₂ in (1.1), we get ! ) , ; ( ₁ ₂ ₁ ₂ 0 n u x z z M n n n    



  2 1 2 1 2 1 ₍₁ ₎ ) 1 ( z z z z x u u           ! ) , ; ( ! ) , ; ( ₂ ₂ 0 1 1 0 p u x z M n u x z M p p p n n n  



     . ! ) , ; ( ) , ; ( ₁ ₁ ₂ ₂ 0 0 n p p n n p n u n x z M x z M p n _           



From the coefficients of u on the both sides of the n

last equality, one can get the desired result. ∎

2. GENERATING FUNCTIONS FOR THE

MEİXNER POLYNOMIALS M_n(z;,x)

We study a number of families of bilinear and bilateral generating functions for the Meixner polynomials M_n(z;,x) which are generated by (1.1) and given by (1.2) using the similar method considered in (see [5], [10]-[16], [19], [21]-[25]). We begin by stating the following theorem.

Theorem 2.1. Corresponding to an identically

non-vanishing function _(s₁,...,s_k) of k complex variables s ,...,₁ s_k (k_N) and of complex order, let ) , , 0 ( ) ,..., ( : ) ; ,..., ( 1 0 1 , C       _ 



        r r k r r r k a s s a s s and





 





. )! ( ) ,..., ( , ; : ; ,..., ; , ; 1 / 0 1 , , pr n s s x z M a s s x z r k r pr n r p n r k p n       



       

Then, forp_N, we have

) ; ,..., ( ) 1 ( ) 1 ( ; ,..., ; , ; 1 , 1 , , 0         k z z n p k p n n s s x u u u u s s x z               



(2.1)

provided that each member of (2.1) exists.

Proof Let S denote the first member of the

assertion (2.1). Then,





. )! ( ) ,..., ( , ; ₁ ] / [ 0 0 n pr u s s x z M a S pr n r k pr n r p n r n r _   _ _    



 _ _  Replacing n by npk, we get

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



! ) ,..., ( , ; ₁ 0 0 n u s s x z M a S n r k n r r n r   ___ 



    ) ; ,..., ( ) 1 ( ) 1 (  z z __,_ s₁ s_k  x u u      

which completes the proof. ∎ By using a similar way, we can write the next result. Theorem 2.2. Corresponding to an identically

non-vanishing function _(s₁,...,s_r) of r complex variables s ,...,₁ s_r (r_N) and of complex order , let ) ; ,..., ; , ; ( ₁ , , 1 2 1 2   z z x s sr p n _ _  N) , , , , 0 ( ) ,..., ( ) , 2 1 ; 2 1 ( : 1 ] [ / 0        _  



p n a s s x z z M a i i i pi n i p n i r C      _ _

and the notation



n / p



means the greatest integer less than or equal n / p .

Then, for pN, we have

) , ; ( 1 1 ] [ 0 0 / x z M pl k pl n a_l _n _k p k l n k          



  l r l pl k z x s s M ( ₂;₂, )___ ( ₁,..., )  _ ) ; ,..., ; , ; ( ₁ ₂ ₁ ₂ ₁ , ,     r p n s s x z z     (2.2)

Proof Let T denote the first member of the assertion (2.2). Then, upon substituting for the polynomials

) , ;

(z₁ z₂ ₁ ₂ x

M_n    from the (1.5) into the left-hand side of (2.2), we obtain

  ) , ; ( ₁ ₁ / 0 0 x z M k pl n a T _l _n _k _pl pl n k p n l           



   l r s l k z x s M ( ₂;₂, )___ ( ₁,..., )    ) , ; ( 1 1 0 / 0 x z M k pl n a _n _k _pl pl n k p n l l             



l l k z x s sr M ( ₂;₂, )___ ( ₁,..., ) 

).

; ,..., ; , ; ( ₁ ₂ ₁ ₂ ₁ , ,     z z x s sr p n _ _   ∎

Theorem 2.3. Corresponding to an identically

non-vanishing function _(s₁,...,s_r)of r complex variables s ,...,1 s_r (rN) and of complex order , let



z x s s_r t



p q,, ;, ; 1,..., ; 





) , 0 ( )! ( ) ,..., ( , ; : ₁ 0 C      _ _ 



  _ n n r pn qn m n n a nq t s s x z M a and   . ) ,..., ( : ) ; ,..., ( 1 / 0 1 , k r pk k q n k r p s s a qk n n s s                



Then, for p_N, we have





! , ( 1,..., ; ) , ; 0 n n t p s sr M_n _m z x n  _   



 z x t t) z m( ) (1    1                      q t t s s t t x z q p r 1 ; ,..., ; 1 , ; , ,  1   (2.3)

Proof Let  denote the first member of the assertion (2.3) of Theorem 2.3. Then,



z x



M_n _m n , ; 0    



    . ! ) ,..., ( ₁ / 0 n n t k k r q n k s s a qk n n pk           





Replacing n by nqk and then using (1.3), we might write that



z x



M n qk n T _n _m _qk k n , ; 0 0              



)! ( ) ,..., ( ₁ qk n qk n t s s a_k _pk _r k     __  )! ( ) ( ) ,..., ( ) 1 , ; ( ) 1 ( ) 1 ( 1 0 kq t s s t t x z M x t t a k q pk q m z qk m z k k r k                       



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Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 21 (6), 1454~1462, 2017 z m z x t t) (1 ) 1 (                         q p q t t s s t t x z _r 1 ; ,..., ; 1 , ; ₁ , ,  

which completes the proof. ∎

3. SPECIAL CASES

We can give many applications of our teorems obtained in the previous section with help of appropriate choices of the multivariable functions

) ,..., (s₁ s_k r    r_N₀, k_N, in terms of simpler function of one and more variables.

For example, if we k1, s₁x and

) , ( ) (x g(s) _r x r        

in Theorem 2.1, where the generalized Cesàro polynomials g(_ns)(,x) [5], generated by         



 ( , ) (1 ) (1 ) 1 ) ( 0 xt t t x g_ns n s n (3.1) The following result which provides a class of bilateral generating functions for the generalized Cesàro polynomials g_n(s)(,x) and the Meixner polynomials. Corollary 3.1. If



 C



 



 _        _ _   , , 0 ) , ( : ) ; , ( ( ) , 0 r r s r r a x g a x r then, we have





)! ( ) , ( , ; ( ) ] / [ 0 0 n pr n u pr u r x g x z M a_r _n _pr s _r r p n n   



      _ _ ) ; , ( ) ( ) (1  1 z ,  x  z x u u       (3.2)

Remark Using the generating relation (3.1) for the

generalized Cesàro polynomials g_n(s)(,x) and getting a_r 1, 0,  1 in Corollary 3.1, we find

that





. ) ( ) ( ) ( ) ( )! ( ) ( 1 1 1 1 , , ; 1 ) ( / ] [ 0 0                       



x x u u x x z s r s r pr n p n r n z z pr n pr n u g M If we set r1, s₁  z₃ and ) , ; ( ) (z₃ _i z₃ ₃ y i M       

in Theorem 2.2, we have the following bilinear generating functions for the Meixner polynomials.

Corollary 3.2. If ) ; , ; ; , ; ( ₁ ₂ ₁ ₂ ₃ ₃ , ,     p z z x z y n    i i pi n i i y z M x z z M a p n     , ) _ _ ( ; , ) ; ( : ₁ ₂ ₁ ₂ ₃ ₃ ] / [ 0     



 ) , , 0 (ai   _C then, we have   ) , ; ( ₁ ₁ / 0 0 x z M k pl n a_l _n _k _pl pl n k p n l          



   l l k z x M z y M ( ₂;₂, ) ___ ( ₃;₃, )  ) ; , ; ; , ; ( 1 2 1 2 3 3 , ,      p z z x z y n _ _   (3.4)

Remark Using (1.5) and taking _,

      l n a_l , 1   x y ,p1,  0,  1 in Corollary 3.2, we have ) , ; ( 1 1 0 0 x z M l n k l n l k n l n k n l                 



 ) , ; ( ) , ; (z₂ ₂ x M z₃ ₃ x M_k__l  _l   ). , ; (z₁ z₂ z₃ ₁ ₂ ₃ x M_n       If we set r 1, s₁ u and ___pn(u)P___pn(u) in Theorem 2.3, where the Legendre polynomials

) (x P_n is generated by [1], 2 0 1 2 1 ) ( t xt t x P_n n n   



  ,

we get a family of the bilateral generating functions for the Legendre polynomials and the Meixner

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polynomials as follows: Corollary 3.3. If







0, N0



)! ( ) ( ) , ; ( : ; ; , ; 0 , ,       



 m a nq t u P x z M a t u x z n n pn n n p q qn m     and   k pk k q n k u P a qk n n u p   _ ( ; ): _ ( ) / 0 ,         



where n,pN, then we have

! ) ; ( ) , ; ( _, 0 n t u x z M n n m p n    _  



                          q t t u t t x z x t t p q z m z 1 ; ; 1 , ; ) 1 ( ) 1 ( , ,    (3.5)

Furthermore, for each a_k (k_N₀), if the multivariable functions ___k(s₁,...,s_r), rN , are expressed as an appropriate product of a lot of simpler functions, the assertions of Theorem 2.1, Theorem 2.2, Theorem 2.3 can be applied in order to derive various families of multilinear and multilateral generating functions for the family of the Meixner polynomials given explicitly by (1.2).

4. FURTHER CONSEQUENCES

In this part we give some special properties for the Meixner polynomials M_n(z;,x) given by (1.2). Theorem 4.1. The Meixner polynomials

) , ; (z x

M_n  have the following integral

representation: ) ( ) ( 1 ) , ; ( z z x z M_n        2 1 1 2 1 1 2 1 ) 2 1 ( 0 0 u u z z n d d u u x u u u u e _          



     

where Re(z)0, Re(z)0.

Proof If we use the identity



Re( ) 0



, ) ( 1 1 0       



e t dt v v a v at v

on the left-hand side of the generating function (1.1), we have ! ) , ; ( 0 n t x z M n n n 



  2 2 2 0 1 1 1 0 1 ) 1 ( 1 ) 1 ( ) ( 1 ) ( 1 du u e z du u e z z u x t z u t          



_ _     

 

2 1 2 1 0 0 0 1 1 2 1 ) 2 1 ( ! ) ( ) ( 1 du du u u t n e z z z z n n x u u u u n       





 _        

From the coefficents of t on the both sides of the n

last equality, one can get the desired result. ∎ We now focus on some miscellaneous recurrence relations of the Meixner polynomials. By differentiating each member of the generating function relation (1.1) with regard to x and using

), , ( ) , ( 0 0 0 0 k n k A n k A n k n k n  



      

we the following differential recurrence relation for the Meixner polynomials have been obtained:

. 1 ), , ; ( ) ( ) , ; ( 1 ₁ 1 0 2      _ _  



n x z M x m n x z x z M _m m _n _m n m n  

Besides, by taking derivative each member of the generating function relation (1.1) with regard to t , we

have the following another recurrence relation for these polynomials: ). , ; ( ) , ; ( ) ( ) , ; ( 0 0 1 x z M x x z x z M z x z M p n p n p m n n m n          



  

5. THE GENERALIZED LAURICELLA FUNCTIONS

Now, we derive various families of bilateral generating functions for the generalized Lauricella (or the Srivastava-Daoust) functions and the Meixner polynomials.

Theorem 5.1. The following bilateral generating function holds true:

! ) ,..., ; ( ) , ; ( ₁ ₂ 0 n t u u u x z M n k n n n  



 

(7)

Sakarya Üniversitesi Fen Bilimleri Enstitüsü Dergisi, 21 (6), 1454~1462, 2017 z z x t t) ( ) (1   1      ) 1 ( 1 ) 1 0 ,..., 2 , , 1 (( )) ) ( ) ( )) (( ₍ p m p m p m k m m p m d z z b      









, ! ... ! ! ! ) ( ,..., ), ,..., ( 2 2 1 1 2 1 2 1 1 1 k m k m p t x t u m t t u k k m u m u p m m m m m f k           where _n(u₁;u₂,...,u_k) is given in (1.4).

Proof With the help of the relationship (1.3), it can be

easily observed that

! ) ,..., ; ( ) , ; ( 1 2 0 n t u u u x z M n k n n n  



     1 )) (( 1 )) (( 1 ) ( 0 0 0 ₁ ₂,..., ) , ; ( m d m b m n k m m n m n n x z M 



     





n k m k m k k n t m u m u m m m m f k ! ! ... ! ,..., ), ,..., ( 1 1 2 1 1  



k k



m m m m m m m m m f d b k ,..., ), ..., , ( )) (( )) (( 2 1 0 ,..., , ₁ 1 2 1  



    ) 1 , ; ( ) 1 ( ) 1 ( ! ... ! ! ) ( 1 1 2 1 2 2 1 1 t t x z M x t t m u m u m t u m z m z k m k m m k        _    _ 1 1 1 2 1 (( )) )) (( ) 1 , ; ( ) 1 ( ) 1 ( 0 ,..., , m m m m m m z z d b t t x z M x t t k     



   





! ... ! ! ) ( ,..., ), ,..., ( 2 2 1 1 2 1 2 1 1 k m k m m t t u k k m u m u m m m m m f k     z z x t t) (1 ) 1 (           p m p m p m m m p m d z z b k       



1 1 1 2 1 )) (( ) ( ) ( )) (( 0 ,..., , ,





! ... ! ! ! ) ( ,..., ), ,..., ( 2 2 1 2 1 2 1 1 1 1 k m k m k k m u m u p m m m m m f k p m t x t u t t u _          ∎ By appropriately choosing the multiple sequence

) ,..., ,

(m₁ m₂ m_s

 in Theorem 5.1, we get a number of interesting results as follows which give bilateral generating functions for the generalized Lauricella functions and the Meixner polynomials.

I. By letting



f(m1,...,mk),m2,...,mk



 ) ( ... ) 1 ( 1 ) ( ... ) 1 ( 1 ) ( ) ( 1 1 k j k m j m k j k m j m j E j j j c a A        



   ) ( ) ( ) 2 ( 2 ) 2 ( 2 ) ( ) ( ... ) ( ) ( ) ( 1 ) ( 1 ) 2 ( 1 ) 2 ( 1 ) ( ) ( ) 2 ( ) 2 ( k j k m k j k m j m j m k j D j k j B j j D j j B j d b d b k k    



    

in Theorem 5.1, we obtain Corollary 5.1 below.

Corollary 5.1. The following bilateral generating function holds true:

) ( ;...; ) 2 ( ; 1 : ) ( ;...; ) 2 ( ; : ) , ; ( 0 k B B B A k D D D E F x z M_n n 



  









      ]; : ) [( ]; : ) [( ]; : ) [( ]; : ) [( ], 1 : [ : ,..., : ) ( : ,..., : ) ( ) 2 ( ) 2 ( ) 2 ( ) 2 ( ) ( ) 1 ( ) ( ) 1 (         d b d b n c a k k _k n k k k k t u u u d b      ,..., , ]; : ) [( ]; : ) [( ...; ...; 2 1 ) ( ) ( ) ( ) (   ) ( ;...; ) 2 ( ; 1 ; 1 : ) ( ;...; ) 2 ( ; 0 ; 0 : ) 1 ( ) 1 ( A B B Bk_k D D D E F x t t z z      









         ]; : ) [( ]; : ) [( ]; : ) [( ]; 1 : [ ], 1 : [ : ,..., : ) ( : ,..., : ) ( ) 2 ( ) 2 ( ) 2 ( ) 2 ( ) 1 ( ) 1 ( ) 1 ( ) 1 (         d b d z z f e k k

 

      _  x t k t u t t u k k k k u u d b ,..., , ), ( ]; : ) [( ]; : ) [( ...; ...; 2 1 ) ( ) ( ) ( ) ( 1 1  

where the coefficients e_j, f_j, (_jk) and (k_j ) are given by ) ( ) 1 ( B A j A A j b a e A j j j _ _ _        ) ( ) 1 ( D E j E E j d c f E j j j _ _ _       

(8)

                              ) 1 2 ; ( 0 ) 2 1 ; ( ) 1 2 ; 1 ( ) 2 1 ; 1 ( ) 1 ( ) 1 ( ) ( k r B A j A r B A j A k r A j r A j A j r j j r j _    and                               ) 1 2 ; ( 0 ) 2 1 ; ( ) 1 2 ; 1 ( ) 2 1 ; 1 ( ) 1 ( ) 1 ( ) ( k r D E j E r D E j E k r E j r E j E j r j j r j _    respectively. II. Upon setting





k m m m m m m k k k k c c b b a m m m m f k k ) ...( ) ( ) ...( ) ( ) ( ,..., ), ,..., ( 1 2 1 1 2 ... 2 1    

and  0 (that is,1...B 1...D 0) in Theorem 5.1, we get the following result.





n k k k k A n n t u u u c c b b n a F x z M ( ; , ) , , 2,..., ; 1,..., ; 1, 2,..., ) ( 0 



   1 ;...; 1 ; 1 ; 1 : 1 1 ;...; 1 ; 0 ; 0 : 1 ) 1 ( ) 1 ( F x t t z  z   











 







          ; 1 : ; 1 : ; ; 1 : ; ; 1 : : ,..., : ) ( : 1 ,..., 1 : ) ( 2 2 ) 1 ( ) 1 ( 1 c b z z c a k   







: 1



;( ),

 

, ,..., ; 1 : ...; ...; 2 1 1 1       _  x t k t u t t u k k u u c b

where the coefficients ( s)

are given by





         . 1 2 , 0 ) 2 1 ( , 1 ) ( k s s s  III. If we put



f(m1,...,mk),m2,...,mk



 k m m k m m k m m c a a a a k k      ... 1 2 2 ) ( ) ...( ) ( ) ...( ) ( 2( 1) ) 1 ( 2 ) 1 ( 1 ) 1 ( 1

and B1, b₁ b, ₁ 1and  0 in Theorem 5.1,

we get Corollary 5.3 below.

) , ; ( 0 x z M_n n 



 





n k k k k B n a a b a a c u u u t F , ,..., , , ,..., ; ; 1, 2,..., ) 1 ( 2 ) 1 ( 2 ) 1 ( 1 ) 1 ( 1 ) (     2 ;...; 2 ; 1 ; 1 : 1 0 ;...; 0 ; 0 ; 0 : 1 ) 1 ( ) 1 ( F x t t z  z   

 











 







           ; ; 1 : ; 1 : ; ; 1 : : 1 ,..., 1 : ) ( : ,..., : (1) ( 1) z z a(1) c b   k 





 

, ,..., , ), ( ; ; 1 : ...; ...; 2 1 ) 1 ( 1 1           k t x t u t t u k u u a

where the coefficients ( s) are given by









         . 1 2 , 0 2 1 , 1 ) ( k s s s  IV. By letting





k m m k m m k m m c b b a m m m m f _k _k s       ... 1 2 ... 1 ) ( ) ...( ) ( ) ( ,..., ), ,..., ( ₁ ₂ 2

an   0, in Theorem 5.1, we get the following result.





n k k k D n n t u u u c b b n a F x z M ( ; , ) ( ) , , ₂,..., ; ; ₁, ₂,..., 0 



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