The (exponential) multipartitional polynomials and polynomial sequences of multinomial type, Part I

ARAB JOURNAL OF MATHEMATICAL SCIENCES

Arab J Math Sci xxx(xx) (2013), xxx–xxx

The (exponential) multipartitional polynomials and polynomial sequences of multinomial type, Part I MILOUD MIHOUBI 1, HACE`NE BELBACHIR *,2 University of Science and Technology Houari Boumediene USTHB, Faculty of Mathematics, P.B. 32 El Alia, 16111 Algiers, Algeria

Abstract. We establish some formulas relating multipartitional polynomials to multinomial polynomials. They appear, respectively, as a natural extension of Bell polynomials and of polynomials of binomial type. Our results are illustrated by some comprehensive examples. Keywords: Multipartitional polynomials; Polynomial sequences of multinomial type; Bell polynomials

1. INTRODUCTION Recently, Mihoubi [4,5] studies the connection between Bell polynomials and binomial type sequences and deduces identities for complete and partial Bell polynomials. As an extension of our previous results on bipartitional polynomials, see [1,6], we establish some connections between multipartitional polynomials and polynomials of multinomial type. They appear, respectively, as a natural extension of Bell polynomials and the polynomials of binomial type. Let us introduce some definitions and notations. We define the complete (exponential) multipartitional polynomial An1 ;...;nr in the variables x0;...;0;1 ; . . . ; xn1 ;...;nr as

* Corresponding author. Tel.: +213 771425883; fax: +213 21276446. E-mail addresses: [email protected], [email protected] (M. Mihoubi), [email protected], [email protected] (H. Belbachir). 1 The research is supported by IFORCE Laboratory of USTHB University. 2 The research is supported by IFORCE Laboratory of USTHB University and CMEP Tassili bilateral Algerian-French program 09MDU765. Peer review under responsibility of King Saud University.

Production and hosting by Elsevier 1319-5166 ª 2013 King Saud University. Production and hosting by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ajmsc.2012.12.001 Please cite this article in press as: M Mihoubi, H Belbachir. The (exponential) multipartitional polynomials and polynomial sequences of multinomial type, Part I. Arab J Math Sci (2013), http://dx.doi.org/10.1016/j.ajmsc.2012.12.001

2

M. Mihoubi, H. Belbachir

An1 ;...;nr ðx0;...;0;1 ; x0;...;0;2 ; . . . ; xn1 ;...;nr Þ :¼

X

n1 !    nr ! k0;...;0;1 !k0;...;0;2 !    kn1 ;...;nr !  kn1 ;...;nr  x k0;...;0;1 xn1 ;...;nr 0;...;0;1   ; ð1Þ 0!    0!1! n1 !    nr !

where the summation is extended over all partitions of the multipartite number (n1 ; . . . ; nr ), that is, over all nonnegative integers (k0;...;0;1 ; . . . ; kn1 ;...;nr ) solution of the equations nr X



n1 X

ir ¼0

ij ki1 ;...;ir ¼ nj ;

j ¼ 1; . . . ; r; with the convention k0;...;0 ¼ 0:

ð2Þ

i1 ¼0

Also, for a given k integer, we define the partial (exponential) multipartitional polynomial of degree k : Bn1 ;...;nr ;k in the variables x0;...;0;1 ; . . . ; xn1 ;...;nr as the sum X n1 !    nr ! Bn1 ;...;nr ;k ðx0;...;0;1 ; x0;...;0;2 ; . . . ; xn1 ;...;nr Þ :¼ k0;...;0;1 !k0;...;0;2 ! . . . kn1 ;...;nr !  kn1 ;...;nr  x k0;...;0;1 xn1 ;...;nr 0;...;0;1   ;ð3Þ 0!    0!1! n1 !    nr ! where the summation is extended over all partitions of the multipartite number (n1 ; . . . ; nr ) into k parts, that is, over all nonnegative integers (k0;...;0;1 ; . . . ; kn1 ;...;nr ) solution of the equations nr X



ir ¼0 nr X

n1 X ij ki1 ;...;ir ¼ nj ;

j ¼ 1; . . . ; r;

i1 ¼0



ir ¼0

ð4Þ

n1 X ki1 ;...;ir ¼ k: i1 ¼0

These polynomials generalize the partial and complete Bell polynomials, see [2,3,7,8], and for other recent results, see [4,5]. Some properties can be deduced from the above definitions, thus: for all real numbers a, b, c we have An1 ;...;nr ðar x0;...;0;1 ; . . . ; an11    anr r xn1 ;...;nr Þ ¼ an11    anr r An1 ;::;nr ðx0;...;0;1 ; . . . ; xn1 ;...;nr Þ;

ð5Þ

Bn1 ;...;nr ;k ðbar x0;...;0;1 ; . . . ; ban11    anr r xn1 ;...;nr Þ ¼ bk an11    anr r Bn1 ;...;nr ;k ðx0;...;0;1 ; . . . ; xn1 ;...;nr Þ: ð6Þ The exponential generating functions for An1 ;...;nr and Bn1 ;...;nr ;k are given by ! X X tn1 tnr t i1 t ir An1 ;...;nr ðx0;...;0;1 ; . . . ; xn1 ;...;nr Þ 1    r ¼ exp xi1 ;...;ir 1    r ; n1 ! nr ! i1 ! ir ! n1 ;...;nr P0 i1 þþir P1 X

n1

tn1 tnr 1 Bn1 ;...;nr ;k ðx0;...;0;1 ; . . . ; xn1 ;...;nr Þ 1    r ¼ n1 ! nr ! k! þþnr Pk

X

t i1 t ir xi1 ;...;ir 1    r i1 ! ir ! i1 þþir P1

ð7Þ

!k : ð8Þ

The polynomials of multinomial type (fn1 ;...;nr ðxÞ) have the following property: f0;...;0 ðxÞ :¼ 1 and

Please cite this article in press as: M Mihoubi, H Belbachir. The (exponential) multipartitional polynomials and polynomial sequences of multinomial type, Part I. Arab J Math Sci (2013), http://dx.doi.org/10.1016/j.ajmsc.2012.12.001

Multipartitional polynomials and 4 polynomial sequences of multinomial type, Part I

!x X ti11 tirr tn1 tnr fi1 ;...;ir ð1Þ    ¼ fn1 ;...;nr ðxÞ 1    r : i1 ! ir ! n1 ! nr ! i1 ;...;ir P0 n1 þþnr P1 X

3

ð9Þ

We use the following notations tn tn11 tnr tn ¼ tn11    tnr r and ¼  r ; n! n1 ! nr ! a ¼ ða1 ; . . . ; ar Þ and

1 ¼ ð1; . . . ; 1Þ;

jnj ¼ n1 þ    þ nr ; n! ¼ n!    nr !; a  b ¼ a1 b1 þ    þ ar br ; a þ b ¼ ða1 þ b1 ; . . . ; ar þ br Þ; ka ¼ ðka1 ; . . . ; kar Þ; ða P bÞ () ða1 P b1 ; . . . ; ar P br Þ; ða > bÞ () ða1 > b1 ; . . . ; ar > br Þ; Dt ¼ Dz¼0

@ @  ; @t1 @tr  d ¼  ; dz z¼0

  x k

81 > < k! xðx  1Þ    ðx  k þ 1Þ if k P 1; ¼ 1 if k ¼ 0 ; > : 0 otherwise

  a i

p

 ¼

a1 i1



 

ar ir ;

x 2 R;

 a1 ; . . . ; ar are real numbers;

Bn;k ðxi Þ ¼ Bn1 ;...;nr ;k ðx0;...;0;1 ; . . . ; xn1 ;...;nr Þ; An ðxi Þ ¼ An1 ;...;nr ðx0;...;0;1 ; . . . ; xn1 ;...;nr Þ;

n P 0; k P 0 and 0 otherwise; n P 0 and 0 otherwise;

Bm;n;k ðxi;j Þ ¼ Bm;n;k ðx0;1 ; x1;0 ; . . . ; xm;n Þ; Am;n ðxi;j Þ ¼ Am;n ðx0;1 ; x1;0 ; . . . ; xm;n Þ; Bm;n;p;k ðxi;j;l Þ ¼ Bm;n;p;k ðx0;0;1 ; x0;1;0 ; . . . ; xm;n;p Þ; Am;n;p ðxi;j;l Þ ¼ Am;n;p ðx0;0;1 ; x0;1;0 ; . . . ; xm;n;p Þ; fn ðxÞ ¼ fn1 ;...;nr ðxÞ

if n P 0; f0 ðxÞ ¼ 1 and 0 otherwise;

Please cite this article in press as: M Mihoubi, H Belbachir. The (exponential) multipartitional polynomials and polynomial sequences of multinomial type, Part I. Arab J Math Sci (2013), http://dx.doi.org/10.1016/j.ajmsc.2012.12.001

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M. Mihoubi, H. Belbachir

es ¼ ð0; . . . ; 0; 1; 0; . . . ; 0Þ s-th position. Let S ¼ ðs1 ; . . . ; sr Þ be a vector with si 2 f0; 1g such that j S jP 1. Let N be the set of nonnegative integers, N ¼ N  f0g and R be the set of real numbers. 2. IDENTITIES

ON MULTIPARTITIONAL POLYNOMIALS

Theorem 1. Let ðxn Þ be a sequence of real numbers. Then for j ¼ 1; . . . ; r, we have X n  i

i

X n  i

i

i

ð10Þ

ij xi Bni;k1 ðxj Þ ¼ nj Bn;k ðxj Þ;

ð11Þ

ij xi Ani ðxj Þ ¼ nj An ðxj Þ:

ð12Þ

p

X n  i

xi Bni;k1 ðxj Þ ¼ kBn;k ðxj Þ;

p

p

Proof. From (8) we have X d tn 1 d X ti Bn;k ðxi Þ ¼ xi dxj n! k! dxj jijP1 i! jnjPk

!k

! X n tm tj tn ¼ Bm;k1 ðxi Þ ¼ Bnj;k1 ðxi Þ : m! j! jnjPjjjþk1 j p n! jmjPk1 X

Then d Bn;k ðxi Þ ¼ dxj

  n j

Bnj;k1 ðxi Þ:

ð13Þ

p

Now, for a1 ¼    ¼ ar ¼ 1 in (6), use (13) and take the derivatives with respect to b of the two sides of (6), this gives Identity (10). We obtain similarly Identity (11). Identity (12) can be deduced by summing the two sides of Identity (11) over all values of k. h  r Theorem 2. P Letj!ðxn Þ be a sequence of real numbers. For given d ¼ ðd1 ; . . . ; dr Þ 2 ðN Þ and yj :¼ d:i¼j i! xi we have

X n! d:n¼n

n!

X n! d:n¼n

n!

Bn;k ðxj Þ ¼ Bn;k ðyj Þ;

ð14Þ

An ðxj Þ ¼ An ðyj Þ;

ð15Þ

Bnes ;k ðxj Þ ¼ Bn;k ðxjes Þ; Anes ðxj Þ ¼ An ðxjes Þ;

s ¼ 1; . . . ; r; s ¼ 1; . . . ; r:

ð16Þ ð17Þ

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Multipartitional polynomials and 4 polynomial sequences of multinomial type, Part I

5

Proof. From (8), setting t1 ¼ td1 ; . . . ; tr ¼ tdr , we obtain !k X 1 X ti tn X tn X n! xi ¼ Bn;k ðxj Þ ¼ Bn;k ðxj Þ; k! jijP1 i! n! nPk n! d:n¼n n! jnjPk 1 X ti xi k! jijP1 i!

!k

1 jX 1 X t j! ¼ xi k! j¼1 j! d:i¼j i!

!k ¼

X tn nPk

n!

Bn;k

X j! d:i¼j

i!

! xi ;

which give Identity (14). For ti ¼ 0, i ¼ 1; . . . ; r, i – s, ts ¼ t in (8), we get !k X 1 X ti tn X tn xies ¼ Bnes ;k ðxi Þ ¼ Bn;k ðxjes Þ ; k! iP1 i! n! nPk n! nPk which gives, by identification, Identity (16). The sum over all possible values of k in the two sides of each of Identities (14) and (16), gives Identities (15) and (17) respectively. h Theorem 3. Let ðxn Þ be a sequence of real numbers. Then Bn;k ðxjjj Þ ¼ Bjnj;k ðxj Þ;

ð18Þ

An ðxjjj Þ ¼ Ajnj ðxj Þ;

ð19Þ



        jnj þ k j jSj1 n þ kS xjjj ¼ ðk!Þ Bjnjþk;k jxjþjSj1 : BnþkS;k jnj S p n p

ð20Þ

Proof. By (8), we have !k !k m X X 1 X ti 1 X xj ðjtjÞ tn j xjij ¼ ðjtjÞ ¼ Bm;k ðxj Þ ¼ Bjnj;k ðxj Þ : k! jijP1 i! k! jP1 j! m! n! mPk jnjPk Then, we obtain Identity (18). Identity (19) follows when we sum over all possible values of k the two sides of Identity (18). Identity (20) follows from the following expansion: !k !k     i !k 1 X i ti 1 X X i t tkS X xj X ti xi ¼ xj ¼ Dt j! k! jP1 j! jij¼j i! k! jijP1 S p i! k! jP1 jij¼j S p i! !k !k kS X t xj tkS X ðjtjÞj1 jjSj ¼ ðjtjÞ ¼ jx k! jPjSj ðj  jSjÞ! k! jP1 jþjSj1 j! ¼ tkS

X

Bm;k ðjxjþjSj1 Þ

mPk

¼t

Q

X

ðjmjÞ! tm Bjmjþk;k ðjxjþjSj1 Þ m! ðjmj þ kÞ! mP0 X  n  BjnjðjSj1Þk;k ðjxjþjSj1 Þ tn jSj1   ; ¼ ðk!Þ jnj  ðjSj  1Þk n! nPkS kS p k P kS

using the fact that

X ðm  kÞ! X tm ðjtjÞmk ¼ tkS Bm;k ðjxjþjSj1 Þ m! m! m! mPk jmj¼mk

i ðksi Þ!

¼ ðk!Þ

i

si

. h

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6

M. Mihoubi, H. Belbachir

Corollary 4. For r ¼ 2 and S ¼ ð1; 0Þ; ð0; 1Þ or ð1; 1Þ we get respectively  Bmþk;n;k ðixiþj Þ ¼ 

k  Bmþnþk;k ðjxj Þ; mþnþk k 

Bm;nþk;k ðjxiþj Þ ¼ 



mþk

nþk



k  Bmþnþk;k ðjxj Þ; mþnþk k 

Bmþk;nþk;k ðijxiþj Þ ¼ k!

  nþk mþk k k   Bmþnþk;k ðjxjþ1 Þ: mþnþk k

The two first identities of Corollary 4 are those of Theorem 3 in [6]. Corollary 5. Let ðxn Þ be a sequence of real numbers. Then, for r ¼ 3 and S ¼ ð1; 0; 0Þ, ð0; 1; 0Þ, ð0; 0; 1Þ, ð1; 1; 0Þ, ð1; 0; 1Þ, ð0; 1; 1Þ or ð1; 1; 1Þ, we get respectively  Bmþk;n;p;k ðixiþjþl Þ ¼ 

mþk



k  Bmþnþpþk;k ðjxj Þ; mþnþpþk k 

 nþk k  Bmþnþpþk;k ðjxj Þ; Bm;nþk;p;k ðjxiþjþl Þ ¼  mþnþpþk k 

 pþk k  Bmþnþpþk;k ðjxj Þ; Bm;n;pþk;k ðlxiþjþl Þ ¼  mþnþpþk k 

mþk

Bmþk;nþk;p;k ðijxiþjþl Þ ¼ k! 



nþk



k k  Bmþnþpþk;k ðjxjþ1 Þ; mþnþpþk k

Please cite this article in press as: M Mihoubi, H Belbachir. The (exponential) multipartitional polynomials and polynomial sequences of multinomial type, Part I. Arab J Math Sci (2013), http://dx.doi.org/10.1016/j.ajmsc.2012.12.001

Multipartitional polynomials and 4 polynomial sequences of multinomial type, Part I



mþk

Bmþk;n;pþk;k ðilxiþjþl Þ ¼ k! 



pþk

7



k k  Bmþnþpþk;k ðjxjþ1 Þ; mþnþpþk k



  pþk nþk k k  Bmþnþpþk;k ðjxjþ1 Þ; Bm;nþk;pþk;k ðjlxiþjþl Þ ¼ k!  mþnþpþk k  Bmþk;nþk;pþk;k ðijlxiþjþl Þ ¼ ðk!Þ2

mþk k 

3. POLYNOMIALS



nþk



pþk

k k  mþnþpþk k

 Bmþnþpþk;k ðjxjþ2 Þ:

OF MULTINOMIAL TYPE

Theorem 6. Let r P 1 be an integer and a ¼ ða1 ; . . . ; ar Þ 2 Rr . If ðfn ðxÞÞ is of multinomial type, then the sequence ðhn ðxÞÞ given by hn ðxÞ :¼

x fn ða  n þ xÞ anþx

ð21Þ

is of multinomial type. Proof. By induction on r. The theorem is true for r ¼ 1 [4, Proposition 1]. Assume the theorem true for k ¼ 1; . . . ; r. Let ðfðn;nÞ ðxÞÞ be a sequence of multinomial type with ðn; nÞ ¼ ðn1 ; . . . ; nr ; nÞ, then by (9), we get ! !x X X X X tnrþ1 tn tn tnrþ1 ti tirþ1 fn;n ðxÞ ¼ fn;n ðxÞ ¼ fi;i ð1Þ n! n! nP0;nP0 n! n! i! i! nP0 nP0 iP0;iP0 ! !x X X ti ti ¼ fi;i ð1Þ rþ1 : i! i! iP0 iP0 If we set un ðxÞ ¼

X

fn;n ðxÞ

nP0

tnrþ1 ; n!

and thus X tn un ðxÞ ¼ n! nP0

X

ti ui ð1Þ i! iP0

!x ;

that is the sequence ðun ðxÞÞ is of multinomial type.

Please cite this article in press as: M Mihoubi, H Belbachir. The (exponential) multipartitional polynomials and polynomial sequences of multinomial type, Part I. Arab J Math Sci (2013), http://dx.doi.org/10.1016/j.ajmsc.2012.12.001

8

M. Mihoubi, H. Belbachir

Under the induction hypothesis, the sequence ðUn ðxÞÞ given by Un ðxÞ ¼ x

X tn un ða  n þ xÞ x ¼ fn;n ða  n þ xÞ rþ1 anþx a  n þ x nP0 n!

is of multinomial type. We then have ! !x X X tn ti Un ðxÞ Ui ð1Þ ; ¼ n! i! nP0 iP0 or equivalently X tn vn ðxÞ rþ1 ¼ n! nP0

X

ti vi ð1Þ rþ1 i! iP0

!x with vn ðxÞ ¼

X fn;n ða  n þ xÞ tn x : a  n þ x n! nP0

This implies that the sequence ðvn ðxÞÞ is of binomial type and the sequence ðVn ðxÞÞ given by X x x tn vn ðarþ1 n þ xÞ ¼ fn;n ða  n þ arþ1 n þ xÞ Vn ðxÞ ¼ arþ1 n þ x a  n þ arþ1 n þ x n! nP0  P1  P1 n ti x is of binomial type too. We have ¼ n¼0 Vn ðxÞ tn!, or equivalently i¼0 Vi ð1Þ i! !x X fi;i ða  i þ arþ1 i þ xÞ ti ti X fn;n ða  n þ arþ1 n þ xtÞ tn tnrþ1 rþ1 x ¼ x : a  i þ arþ1 i þ x i! i! a  n þ arþ1 n þ x n! n! iP0;iP0 nP0;nrþ1 P0 This proves that the sequence ðhn;n ðxÞÞ given by x fn;n ða  n þ arþ1 n þ xÞ hn;n ðxÞ :¼ a  n þ arþ1 n þ x is of multinomial type.

h

From (9), one can infer that if the sequence ðfn ðxÞÞ is of multinomial type, then X n  fn ðx þ yÞ ¼ fi ðxÞfni ðyÞ: ð22Þ i p i ð1Þ

ðrÞ

Theorem 7. Let ðfn ðxÞÞ, ðgn ðxÞÞ; . . . ; ðgn ðxÞÞ be sequences of binomial type. Then the polynomials ðpn ðxÞÞ and ðqn ðxÞÞ defined by pn ðxÞ ¼ fjnj ðxÞ and

qn ðxÞ ¼ gnð1Þ ðxÞ    gðrÞ nr ðxÞ 1

ð23Þ

are of multinomial type. Proof. We have p0 ðxÞ ¼ q0 ðxÞ ¼ 1, X X tn X tn X ðjtjÞk pn ðxÞ ¼ fk ðxÞ ¼ fk ðxÞ ¼ n! kP0 n! kP0 k! nP0 jnj¼k

X

ðjtjÞk fk ð1Þ k! kP0

!x ;

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Multipartitional polynomials and 4 polynomial sequences of multinomial type, Part I

and

! r X X tn Y tnkk ðkÞ qn ðxÞ ¼ g ðxÞ ¼ n! k¼1 n P0 nk nk ! nP0 k

r Y X k¼1

tnkk gðkÞ nk ð1Þ nk ! nk P0

9

!!x :



Remark 8. Let ðfn ðxÞÞ, ðgnð1Þ ðxÞÞ; . . . ; ðgðrÞ n ðxÞÞ be sequences of binomial type. Then, using Theorem 6, the sequences Pn ðxÞ ¼

x fjnj ða  n þ xÞ anþx

and

Qn ðxÞ ¼

r Y x gðiÞ ða  n þ xÞ a  n þ x i¼1 ni

are of multinomial type. 4. MULTIPARTITIONAL

POLYNOMIALS AND MULTINOMIAL TYPE POLYNOMIALS

Roman [8, p. 82], proved that any sequence of binomial type ðfn ðxÞÞ, f0 ðxÞ ¼ 1, is related to the partial Bell polynomials in the form n X fn ðxÞ ¼ Bn;k ðDa¼0 fj ðaÞÞxk ; n P 1: ð24Þ k¼1

Similarly, we will establish that a sequence of multinomial type ðfn ðxÞÞ, and the partial multipartitional polynomials are related in the form given by the following theorem: Theorem 9. Let ðfn ðxÞÞ be a multinomial type sequence. Then, for ŒnŒ P 1, we have fn ðxÞ ¼

jnj X Bn;k ðDa¼0 fi ðaÞÞxk ¼ An ðxDa¼0 fi ðaÞÞ:

ð25Þ

k¼1

Proof. We have !k 1 tn X xk X ti Bn;k ðDa¼0 fi ðaÞÞx ¼ x Bn;k ðDa¼0 fi ðaÞÞ ¼ Da¼0 fi ðaÞ n! k¼0 n! k¼0 k! iP0 i! nP0 k¼0 jnjPk !! ! X X ti ti ¼ exp xDa¼0 fi ðaÞ ¼ exp x Da¼0 fi ðaÞ i! i! iP0 iP0 !a ! !! X X ti ti ¼ exp x ln fi ð1Þ fi ð1Þ ¼ exp xDa¼0 i! i! iP0 iP0 !x X X ti tn fi ð1Þ ¼ fn ðxÞ ; ¼ i! n! iP0 nP0

jnj X tn X

k

1 X

k

X

so, the desired identity follows by identification. h

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10

M. Mihoubi, H. Belbachir

Theorem 10. Let a 2 R, a ¼ ða1 ; . . . ; ar Þ 2 Rr and ðfn ðxÞÞ be a multinomial type sequence. Then for n P kS, we have       i n fiS ða  ði  SÞ þ aÞ fnkS ða  ðn  kSÞ þ akÞ ¼ akðk!ÞjSj1 Bn;k a : a  ðn  kSÞ þ ak S p a  ði  SÞ þ a kS p

ð26Þ

Proof. We have !k    n   X i t 1 X i ti Bn;k fiS ðaÞ ¼ fiS ðaÞ n! k! jijP1 S p i! S p jnjPk !k tkS X ti ¼ fi ðaÞ k! iP0 i! tkS X tn fn ðakÞ k! nP0 n!  X n  tn jSj1 ¼ ðk!Þ fnkS ðakÞ ; n! kS p nPkS

¼

i.e. Bn;k

     i n jSj1 fiS ðaÞ ¼ ðk!Þ fnkS ðakÞ: S p kS p

It suffices to replace ðfn ðxÞÞ by ðhn ðxÞÞ given by (21).

ð27Þ h

Corollary 11. Let ðfn ðxÞÞ be a binomial type sequence. Then, for r ¼ 1, S ¼ ð1Þ, a ¼ a, the polynomials Bn;k ðxi Þ represent the partial Bell polynomials and     n fnk ðaðn  kÞ þ akÞ fi1 ðaðði  1ÞÞ þ aÞ Bn;k ai : ¼ ak aði  1Þ þ a aðn  kÞ þ ak k Corollary 12. Let ðfm;n ðxÞÞ be a trinomial type sequence. Then, for r ¼ 2, S ¼ ð1; 0Þ, a ¼ ða; bÞ, we get     m fmk;n ðaðm  kÞ þ bn þ akÞ fi1;j ðaði  1Þ þ bj þ aÞ Bm;n;k ai ; ¼ ak aði  1Þ þ bj þ a aðm  kÞ þ bn þ ak k for r ¼ 2, S ¼ ð0; 1Þ, a ¼ ða; bÞ, we get     m fm;nk ðam þ bðn  kÞ þ akÞ fi;j1 ðai þ bðj  1Þ þ aÞ Bm;n;k aj ; ¼ ak ai þ bðj  1Þ þ a am þ bðn  kÞ þ ak k and for r ¼ 2, S ¼ ð1; 1Þ, a ¼ ða; bÞ, we get   fi1;j1 ðaði  1Þ þ bðj  1Þ þ aÞ Bm;n;k aij aði  1Þ þ bðj  1Þ þ a    m n fmk;nk ðaðm  kÞ þ bðn  kÞ þ akÞ ¼ ak : aðm  kÞ þ bðn  kÞ þ ak k k

Please cite this article in press as: M Mihoubi, H Belbachir. The (exponential) multipartitional polynomials and polynomial sequences of multinomial type, Part I. Arab J Math Sci (2013), http://dx.doi.org/10.1016/j.ajmsc.2012.12.001

Multipartitional polynomials and 4 polynomial sequences of multinomial type, Part I

11

The identity of Corollary 11 is Proposition 1 in [4] and the identities of Corollary 12 are those of Theorems 10 and 11 in [6]. Corollary 13. Let ðfm;n;p ðxÞÞ be a multinomial type sequence with f0;0;0 ðxÞ ¼ 1. Then, for r ¼ 3, S ¼ ð1; 0; 0Þ, a ¼ ða; b; cÞ, we get  Bm;n;p;k

   m fmk;n;p ðaðm  kÞ þ bn þ cp þ akÞ fi1;j;l ðaði  1Þ þ bj þ cl þ aÞ ai ; ¼ ak aði  1Þ þ bj þ cl þ a aðm  kÞ þ bn þ cp þ ak k

for r ¼ 3, S ¼ ð0; 1; 0Þ, a ¼ ða; b; cÞ, we get     n fm;nk;p ðam þ bðn  kÞ þ cp þ akÞ fi;j1;l ðai þ bðj  1Þ þ cl þ aÞ ¼ ak Bm;n;p;k aj ; ai þ bðj  1Þ þ cl þ a am þ bðn  kÞ þ cp þ ak k for r ¼ 3, S ¼ ð0; 0; 1Þ, a ¼ ða; b; cÞ, we get     p fm;n;pk ðam þ bn þ cðp  kÞ þ akÞ fi;j;l1 ðai þ bj þ cðl  1Þ þ aÞ ¼ ak Bm;n;p;k al ; ai þ bj þ cðl  1Þ þ a am þ bn þ cðp  kÞ þ ak k for r ¼ 3, S ¼ ð1; 1; 0Þ, a ¼ ða; b; cÞ, we get   fi1;j1;l ðaði  1Þ þ bðj  1Þ þ cl þ aÞ Bm;n;p;k aij aði  1Þ þ bðj  1Þ þ cl þ a    n fmk;nk;p ðaðm  kÞ þ bðn  kÞ þ cp þ akÞ m ¼ ak! ; aðm  kÞ þ bðn  kÞ þ cp þ ak k k for r ¼ 3, S ¼ ð1; 0; 1Þ, a ¼ ða; b; cÞ, we get   fi1;j;l1 ðaði  1Þ þ bj þ cðl  1Þ þ aÞ Bm;n;p;k ail aði  1Þ þ bj þ cðl  1Þ þ a    m p fmk;n;pk ðaðm  kÞ þ bn þ cðp  kÞ þ akÞ ¼ ak! ; aðm  kÞ þ bn þ cðp  kÞ þ ak k k for r ¼ 3, S ¼ ð0; 1; 1Þ, a ¼ ða; b; cÞ, we get   fi;j1;l1 ðai þ bðj  1Þ þ cðl  1Þ þ aÞ Bm;n;p;k ajl ai þ bðj  1Þ þ cðl  1Þ þ a    p fm;nk;pk ðam þ bðn  kÞ þ cðp  kÞ þ akÞ n ¼ ak! ; am þ bðn  kÞ þ cðp  kÞ þ ak k k and for r ¼ 3, S ¼ ð1; 1; 1Þ, a ¼ ða; b; cÞ, we get   fi1;j1;l1 ðaði  1Þ þ bðj  1Þ þ cðl  1Þ þ aÞ Bm;n;p;k aijl aði  1Þ þ bðj  1Þ þ cðl  1Þ þ a     m n p fmk;nk;pk ðaðm  kÞ þ bðn  kÞ þ cðp  kÞ þ akÞ : ¼ aðk!Þ2 aðm  kÞ þ bðn  kÞ þ cðp  kÞ þ ak k k k

Please cite this article in press as: M Mihoubi, H Belbachir. The (exponential) multipartitional polynomials and polynomial sequences of multinomial type, Part I. Arab J Math Sci (2013), http://dx.doi.org/10.1016/j.ajmsc.2012.12.001

12

M. Mihoubi, H. Belbachir

Example 14. When fn ðaÞ ¼ ajnj in (26), we get       i n jiSj1 jSj1 jnj1 Bn;k a ða  ði  SÞ þ aÞ ða  ðn  kSÞ þ akÞ ¼ akðk!Þ S p kS p   a1 and when fn ðaÞ ¼ n! , we get n  p      i ða  ði  SÞ þ aÞ1 ði  SÞ! Bn;k a a  ði  SÞ þ a S p iS p     n ða  ðn  kSÞ þ akÞ1 ðn  kSÞ! jSj1 ¼ akðk!Þ : a  ðn  kSÞ þ ak kS p n  kS p Using Theorem 7, Theorem 10 becomes Corollary 15. Let a 2 R, a ¼ ða1 ; . . . ; ar Þ 2 Rr and ðfn ðxÞÞ be a sequence of binomial type. Then, for n P kS, we have       i n fjiSj ða  ði  SÞ þ aÞ fjnkSj ða  ðn  kSÞ þ akÞ jSj1 Bn;k a : ¼ akðk!Þ a  ði  SÞ þ a a  ðn  kSÞ þ ak S p kS p ð28Þ   a Example 16. When fn ðaÞ ¼ n! in (28), one has n      a  ði  SÞ þ a a  ðn  kSÞ þ ak ðjijÞ! ak ðjnjÞ! ¼ Bn;k a a  ði  SÞ þ a k! a  ðn  kSÞ þ ak ji  Sj jn  kSj and when fn ðaÞ ¼ Bn ðaÞ, (the single variable Bell polynomials),       i n BjiSj ða  ði  SÞ þ aÞ BjnkSj ða  ðn  kSÞ þ akÞ ¼ akðk!ÞjSj1 Bn;k a : a  ði  SÞ þ a a  ðn  kSÞ þ ak S p kS p Theorem 17. Let a 2 R, a ¼ ða1 ; . . . ; ar Þ 2 Rr and ðfn ðxÞÞ be a multinomial type sequence. We have   fi ða  iÞ fn ða  n þ aÞ An a : ¼a ai anþa

ð29Þ

Proof. From (25), we infer An ðaDx¼0 fi ðxÞÞ ¼ fn ðaÞ; which gives the desired identity by replacing fn ðaÞ by hn ðaÞ given by (21).

h

Corollary 18. Let ðfn ðxÞÞ be a binomial type sequence. For r ¼ 1, a ¼ a, the polynomials An ðxi Þ represent the complete Bell polynomials, and we have  Am

fi ðaiÞ a ai

 ¼a

fm ðam þ aÞ : am þ a

Please cite this article in press as: M Mihoubi, H Belbachir. The (exponential) multipartitional polynomials and polynomial sequences of multinomial type, Part I. Arab J Math Sci (2013), http://dx.doi.org/10.1016/j.ajmsc.2012.12.001

Multipartitional polynomials and 4 polynomial sequences of multinomial type, Part I

13

Corollary 19. Let ðfm;n ðxÞÞ be a trinomial type sequence. Then, for r ¼ 2, a ¼ ða; bÞ,   fi;j ðai þ bjÞ fm;n ðam þ bn þ aÞ Am;n a ; ¼a ai þ bj am þ bn þ a and for r ¼ 3, a ¼ ða; b; cÞ,   fi;j ðai þ bj þ clÞ fm;n;p ðam þ bn þ cp þ aÞ Am;n;p a : ¼a ai þ bj þ cl am þ bn þ cp þ a The identity of Corollary 18 is Proposition 3 in [5] and the identities of Corollary 19 are those of Theorem 14 in [6]. Example 20. For fn ðaÞ ¼ ajnj in (29), we get An ðaða  iÞjij1Þ ¼  aða  n þ aÞjnj1 ; x1 and for fn ðaÞ ¼ n! , we have   n p    ða  iÞ1 ða  n þ aÞ1 i! n! An a : ¼a ai anþa i n p p Using Theorem 7, Theorem 17 becomes Corollary 21. Let a 2 R, a ¼ ða1 ; . . . ; ar Þ 2 Rr and ðfn ðxÞÞ be a sequence of binomial type. We have   fjij ða  iÞ fjnj ða  n þ aÞ An a : ð30Þ ¼a ai anþa   a Example 22. For fn ðaÞ ¼ n! , we get from (30): n      anþa ðjijÞ! a  i ðjnjÞ! An a ¼a ; ai anþa jij jnj and for fn ðaÞ ¼ Bn ðaÞ,   Bjij ða  iÞ Bjnj ða  n þ aÞ An a : ¼a ai anþa REFERENCES [1] H. Belbachir, M. Mihoubi, The (exponential) bipartitional polynomials and polynomial sequences of trinomial type, Part II, Integers 11 (2011) A29, 12 p. [2] Charalambos A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, A CRC Press Company, Boca Raton, London, New York, Washington, DC, 2001. [3] L. Comtet, Advanced Combinatorics, Reidel, Dordrecht, Netherlands, 1974. [4] M. Mihoubi, Bell polynomials and binomial type sequences, Discrete Math. 308 (2008) 2450–2459. [5] M. Mihoubi, The role of binomial type sequences in determination identities for Bell polynomials. Ars Combin. (2013), in press. Preprint available at: . [6] M. Mihoubi, H. Belbachir, The (exponential) bipartitional polynomials and polynomial sequences of trinomial type, Part I, Integers 11 (2011) A18, 17 p. [7] J. Riordan, Combinatorial Identities, Huntington, New York, 1979. [8] S. Roman, The Umbral Calculus, Academic Press, New York, 1984.

Please cite this article in press as: M Mihoubi, H Belbachir. The (exponential) multipartitional polynomials and polynomial sequences of multinomial type, Part I. Arab J Math Sci (2013), http://dx.doi.org/10.1016/j.ajmsc.2012.12.001