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

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

HACE`NEBELBACHIR*, MILOUDMIHOUBI

University of Science and Technology Houari Boumediene, USTHB, Faculty of Mathematics, P.B. 32, El Alia, 16111 Algiers, Algeria

Received 15 December 2012; revised 29 May 2013; accepted 10 June 2013 Available online 29 June 2013

Abstract. We establish recursiveness properties for multipartitional polynomials and their connection with the derivatives of polynomials of multinomial type. Various com- prehensive examples are illustrated.

Mathematis Subject Classification: 11B83; 05A10; 11B65

Keywords: Multipartitional polynomials; Derivatives of polynomial sequences of multinomial type; Recursiveness relations

1. INTRODUCTION

In a precedent work[6], the authors established identities on multipartitional polyno- mials, generalizing[1]. They give a result concerning the preservation of sequences of multinomial type. These results appear as natural extension of Bell polynomials and of polynomials of binomial type (see for instance [2–4]). Finally, they give some connection between these two concepts.

Let us introduce, as in[6], some definitions and notations.

Form,m1,. . .,mr integers such thatPr

i¼1mi¼m;we define m

m1;. . .;mr

¼

m!

m1!mr! if miP0; i¼1;. . .;r;

0 otherwise:

LetN be the set of nonnegative integers and Rthe set of real numbers. We use, the following notations, fora¼ ða1;. . .;arÞ 2R^r;b¼ ðb1;. . .;brÞ 2R^rwe set

* Corresponding author. Tel.: +213 771425883; fax: +213 21276446.

E-mail addresses:[email protected](H. Belbachir),[email protected](M. Mihoubi).

Peer review under responsibility of King Saud University.

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1319-5166ª2013 Production and hosting by Elsevier B.V. on behalf of King Saud University.

http://dx.doi.org/10.1016/j.ajmsc.2013.06.003

Arab J Math Sci21(1) (2015), 2–14

ARABJOURNAL OF

MATHEMATICALSCIENCES

a.b=a1b1+ +arbr,a+b = (a1+b1,. . .,ar+br),ka= (ka1,. . .,kar), (aPb) () (a1Pb1,. . .,arP br), (a>b) () (a1>b1,. . .,ar>br), 1_ðaPbÞ¼ 1 if aPb;

0 otherwise;

forn¼ ðn1;. . .;n_rÞ 2N^r we set

tⁿ¼tⁿ₁¹ tⁿ_r^r;^t_n!ⁿ¼^t

n1 1 n1! ^t_n^nrr

r!;n! =n1! nr!,ŒnŒ=n1+ +nr,1= (1,. . ., 1), n

i _p ¼ n₁ i₁

n_r

i_r ; n

i;j;. . .;k

p

¼ n₁ i₁;j₁;. . .;k₁

n_r i_r;j_r;. . .;k_r

;

B_n;kðxiÞ ¼B_n1;...;nr;kðx0;...;0;1;. . .;x_n1;...;nrÞ,nP 0,kP0 and =0 otherwise, A_nðxiÞ ¼A_n1;...;nrðx0;...;0;1;. . .;x_n1;...;nrÞ,nP0 and =0 otherwise,

f_nðxÞ ¼fn₁;...;nrðxÞifnP 0,f₀(x) = 1 and =0 otherwise, for ðm;nÞ 2N² we set Bm,n;k(xi,j) =Bm,n;k(x0,1,x1,0 ,. . .,xm,n),

Am,n(xi,j) =Am,n(x0,1,x1,0,. . .,xm,n) and we set alsoD_z¼0¼_dz^d

z¼0.

The complete (exponential) multipartitional polynomials A_n in the variables (x_i, i„0), are defined by the sum

AnðxiÞ:¼X n!

Yⁿ

i¼0;i–0

k_i! Yⁿ

i¼0;i–0

xi

i!

k_i

; ð1Þ

where the summation is extended over all partitions of the multipartite number n= (n1,. . .,nr), that is, over all nonnegative integers (ki,06i6n) solution of the equations

Xn

i¼0ijk_i¼nj; j¼1;. . .;r; with convention k₀¼0: ð2Þ

The partial (exponential) multipartitional polynomialsBn,kin the variables (xi,i„0), of degreek, are defined by the sum

Bn;kðxiÞ:¼X n!

Qn

i¼0;i–0ðki!Þ Yn

i¼0;i–0

xi

i!

ki

; ð3Þ

where the summation is extended over all partitions of the multipartite number n= (n1,. . .,nr) intokparts, that is, over all nonnegative integers (ki,06i6n) solution of the equations

Xn

i¼0ki¼k; Xn

i¼0ijki¼nj; j¼1;. . .;r; with conventionk0¼0: ð4Þ

Also, we can verify that the exponential generating function forAn1;...;nr is given by X

nP0AnðxjÞtⁿ

n!¼exp X

jijP1xi

tⁱ i!

ð5Þ

and the exponential generating function forBn; k is given by X

jnjPkBn;kðxjÞtⁿ n!¼ 1

k!

X

jijP1x_itⁱ i!

^k

: ð6Þ

The polynomials of multinomial type (f_n(x)) are defined byf₀(x):¼1 and X

iP0fið1Þtⁱ i!

^x

¼X

nP0fnðxÞtⁿ

n!: ð7Þ

In the followingSdesign a vector ofN^rn f0g.

We will use the two following results, given in[6]:

Theorem 1. Let rP1 be an integer anda2R^r. If (f_n(x)) is a multinomial type sequence of polynomials, then the polynomials (h_n(x)) and (p_n(x)) given by

h_nðxÞ:¼ x

a:nþxf_nða:nþxÞ ð8Þ

p_nðxÞ:¼f_jnjðxÞ ð9Þ

are of multinomial type.

Theorem 2. Leta2R;a2R^r and (fn(x)) be a sequence of polynomials of multinomial type. We have

A_n afiða:iÞ a:i

¼afnða:nþaÞ

a:nþa : ð10Þ

In this paper, we start with two light extensions concerning multipartitional polyno- mials given in[6]. The main results of the paper concern the recursiveness properties for multipartitional polynomials and their connections with the derivatives of polynomials of multinomial type. Each result is followed by various applications.

2. IDENTITIES RELATED TO MULTIPARTITIONAL POLYNOMIALS

In[6], Theorems 3 and 4 are established when S2 {0,1}^r,S„0. Here, we extend the results to any vectorS2N^rsuch thatŒSŒP1. These extensions give an other applica- tion illustrated by vectorial multinomial coefficient.

Theorem 3. Let (xn) be a sequence of real numbers. Then jnj þk

jnj

!

B_nþkS;k j

S !

p

x_jjj

!

¼ 1 k!

nþkS n;S;. . .;S

!

p

B_jnjþk;kðjx_jþjSj1Þ: ð11Þ

Proof. Identity(11)follows from the following expansion:

1 k!

P

jijP1

i S _px_i^t_i!^{i k}

¼_k!¹ P

jP1xjP

jij¼j

i S _p

tⁱ i!

^k

¼ ^tkS

k!ðS!Þ^k

P

jP1xjP

jij¼j t^iS ðiSÞ!

k

¼ ^tkS

k!ðS!Þ^k

P

jP1xjþjSj1P

jij¼j1tⁱ i!

k

¼ ^tkS

k!ðS!Þ^k

P

jP1jx_jþjSj1^ðjtjÞ_j!^j1 k

¼ ^tkS

ðS!Þ^k

P

mPkBm;kðjx_jþjSj1Þ^ðjtjÞ_m!^mk

¼_ðS!Þ^tkSk

P

mPk ðmkÞ!

m! Bm;kðjx_jþjSj1ÞP

jmj¼mkt^m m!

¼ ^tkS

ðS!Þ^k

P

mP0 ðjmjÞ!

ðjmjþkÞ!Bjmjþk;kðjx_jþjSj1Þ^t_m!^m

¼_k!ðS!Þ¹ k

P

mP0 n!

ðnkSÞ!

BjnjkjSjþk;kðjxjþjSj1Þ

jnj ðjSj 1Þk k

^t_n!ⁿ

¼_k!¹P

nPkS

n nkS;S;. . .;S

p

BjnjðjSj1Þk;kðjxjþjSj1Þ

jnj ðjSj 1Þk k

^t_n!ⁿ:

Corollary 4. Let (xn) be a sequence of real numbers. Then, for r = 1 andS= u, we get nþk

n

B_nþku;k j u xj

¼ 1 k!

nþku n;u;. . .;u

B_nþk;kðjx_jþu1Þ:

Corollary 5. Let (xn) be a sequence of real numbers. Then, for r = 2 andS= (u,0), we get

mþnþk mþn

Bmþku;n;k

i u xiþj

¼ 1 k!

mþku m;u;. . .;u

Bmþnþk;kðjx_jþu1Þ;

for r = 2 and S= (0,v), we get mþnþk

mþn

B_m;nþkv;k j v x_iþj

¼ 1 k!

nþkv n;v;. . .;v

B_mþnþk;kðjx_jþv1Þ;

for r = 2 and S= (u,v), we get mþnþk

mþn

Bmþku;nþkv;k

i u

j v x_iþj

¼ 1 k!

mþku m;u;. . .;u

nþkv n;v;. . .;v

Bmþnþk;kðjx_jþuþv1Þ:

Theorem 6. Leta2R;a2R^rand (fn(x)) be a multinomial type sequence of polynomi- als. We have

Bn;k a i S _p

f_iSða:ðiSÞ þaÞ a:ðiSÞ þa

¼ a ðk1Þ!

n nkS;S;. . .;S

p

fnkSða:ðnkSÞ þakÞ

a:ðnkSÞ þak : ð12Þ

Proof. We have P

jnjPkBn;k

i

S _pfiSðaÞ

tⁿ

n!¼_k!¹ P

jijP1

i

S _pfiSðaÞ^t_i!ⁱ k

¼ ^tkS

k!ðS!Þ^k

P

iP0fiðaÞ^t_i!^{i k}

¼ ^tkS

k!ðS!Þ^k

P

nP0fnðakÞ^t_n!ⁿ

¼ ¹

k!ðS!Þ^k

P

nPkS n!

ðnkSÞ!fnkSðakÞ^t_n!ⁿ; i.e.

Bn;k

i

S _pfiSðaÞ

¼ 1 k!

n nkS;S;. . .;S

p

fnkSðakÞ: ð13Þ It suffices to replace (fn(x)) by (hn(x)) as in(8). h

Corollary 7. Leta;a2Rand (fn(x)) be a binomial type sequence of polynomials. Then, for r = 1 andS= u, we have

Bn;k a i u

fiuðaðiuÞ þaÞ aðiuÞ þa

¼ a ðk1Þ!

n nku;u;. . .;u

fnkuðaðnkuÞ þakÞ aðnkuÞ þak : Corollary 8. Let a2R;a¼ ða;bÞ 2R² and (fm,n(x)) be a trinomial type sequence of polynomials. Then, for r = 2 andS= (u,0), we have

Bm;n;k a i u

fiu;jðaðiuÞ þbjþaÞ aðiuÞ þbjþa

¼ a ðk1Þ!

m mku;u;. . .;u

f_mku;nðaðmkuÞ þbnþakÞ aðmkuÞ þbnþak ; for r = 2 andS= (0,v), we have

Bm;n;k a j v

f_i;jvðaiþbðjvÞ þaÞ aiþbðjvÞ þa

¼ a ðk1Þ!

n nkv;v;. . .;v

fm;nkvðamþbðnkvÞ þakÞ amþbðnkvÞ þak ; for r = 2 andS= (u,v), we have

Bm;n;k a i u

j v

fiu;jvðaðiuÞ þbðjvÞ þaÞ aðiuÞ þbðjvÞ þa

¼ a ðk1Þ!

m!n!

ðmkuÞ!ðnkvÞ!ðu!v!Þ^k

fmku;nkvðaðmkuÞ þbðnkvÞ þakÞ aðmkuÞ þbðnkvÞ þak :

3. RECURSIVENESS IN MULTIPARTITIONAL POLYNOMIALS

This section is devoted to the twice composition of multipartitional polynomials.

Theorem 9. Let (x_n) be a sequence of real numbers with xS¼1;d2N^r and d2N^I. We have

Bn;k 1_ðiPSÞðt1Þ!d iþ ðt1ÞS i;S;. . .;S ¹

p

Biþðt1ÞS;t

j S _pxj

!

¼1_ðnPkSÞðT1Þ!d ðk1Þ!

nþ ðTkÞS n;S;. . .;S 1

p

B_nþðTkÞS;T j S _px_j

; ð14Þ

where t =dÆ(iS) + d T =dÆ(nkS) + kd.

Proof. Let (fn(x)) be a multinomial type sequence of polynomials with fn(1):¼xn+S. Fora=0 anda= 1 in(12), we get

fnðkÞ

k ¼ ðk1Þ! nþkS n;S;. . .;S 1

p

B_nþkS;k i S _px_i

: ð15Þ

Fora=d anda=d, Formula(12)becomes Bn;k d i

S _p fiSðtÞ

t

¼ d ðk1Þ!

n nkS;S;. . .;S

p

fnkSðTÞ

T : ð16Þ

Therefore, form(15), we obtain fiSðtÞ

t ¼ ðt1Þ! iþ ðt1ÞS iS;S;. . .;S ¹

p

Biþðt1ÞS;t

i S _pxi

and f_nkSðTÞ

T ¼ ðT1Þ! nþ ðTkÞS nkS;S;. . .;S 1

p

B_nþðT1ÞS;T i S _px_i

:

Now, replacing in(16)f_iS(t) andf_nkS(T) by their above expressions, we get Bn;k ðt1Þ!d i

S _p

iþ ðt1ÞS iS;S;. . .;S ¹

p

Biþðt1ÞS;t

i S _pxi

!

¼ðT1Þ!d ðk1Þ!

n nkS;S;. . .;S

p

nþ ðTkÞS nkS;S;. . .;S ¹

p

BnþðTkÞS;T

i S _pxi

:

Then, to obtain(14), it suffices to observe that i

S _p

iþ ðt1ÞS iS;S;. . .;S 1

p

¼1_ðiPSÞ iþ ðt1ÞS i;S;. . .;S 1

p

;

n nkS;S;. . .;S

p

nþ ðTkÞS nkS;S;. . .;S 1

p

¼1_ðnPkSÞ nþ ðTkÞS n;S;. . .;S 1

p

:

Corollary 10. Let (fn(x)) be a binomial type sequence of polynomials and d2N^I. Then, for r = 1,S= u andd¼p2N;and Bn,k(xi) be the partial Bell polynomials, we get

Bn;k 1_ðiPuÞðt11Þ!d iþ ðt11Þu i;u;. . .;u 1

B_{iþðt11Þu;t1} j u xj

!

¼1_ðnPkuÞðT11Þ!d ðk1Þ!

nþ ðT1kÞu n;u;. . .;u 1

B_{nþðT1kÞu;T1} j u xj

;

where t1= p(iu) + d, T1= p(nku) + kd.

Corollary 11. Let p, q, dP1 be nonnegative integers and (fm,n(x)) be a trinomial type sequence of polynomials. Then, for r = 2,d= (p, q) andS= (u, 0), we get

Bm;n;k 1_ðiPuÞðt101Þ!d iþ ðt101Þu i;u;. . .;u 1

B_{iþðt101Þu;j;t10} i⁰ u x_i⁰_;j⁰ !

¼1_ðmPkuÞðT101Þ!d ðk1Þ!

mþ ðT10kÞu m;u;. . .;u 1

B_{mþðT10kÞu;n;T10} i⁰ u x_i⁰_;j⁰

;

where t10= p(iu) + qj + d and T10= p(mku) + qn + kd; for r = 2,d= (p,q) andS= (0,v), we get

Bm;n;k 1_ðjPvÞðt011Þ!d jþ ðt011Þv j;v;. . .;v 1

B_{i;jþðt011Þv;t01} j⁰ v xi⁰;j⁰

!

¼1_ðnPkvÞðT011Þ!d ðk1Þ!

nþ ðT01kÞv n;v;. . .;v 1

B_{m;nþðT01kÞv;T01} j⁰ v xi⁰;j⁰

;

where t01= pi + q(jv) + d and T01= pm + q(nkv) + kd; for r = 2,d= (p,q) andS= (u,v), we get

Bm;n;k 1_ðiPuÞ1_ðjPvÞðt111Þ!d

B_{iþðt111Þu;jþðt111Þv;t11} i⁰ u

j⁰ v x_i⁰_;j⁰

iþ ðt111Þu i;u;. . .;u

jþ ðt111Þv j;v;. . .;v

0 BB B@

1 CC CA

¼1_ðmPkuÞ1_ðnPkvÞðT111Þ!d ðk1Þ!

BmþðT11kÞu;nþðT11kÞv;T11

i⁰ u

j⁰ v xi⁰;j⁰

mþ ðT11kÞu m;u;. . .;u

nþ ðT11kÞv n;v;. . .;v ;

where t11= p(iu) + q(jv) + d and T11= p(mku) + q(nkv) + dk.

Foru= 1 in Corollary 10 we obtain Proposition 4 of[3]and foru=v= 1 in Cor- ollary 11 we obtain Theorems 7 and 10 of[5].

Using some particular cases related to Bell polynomials. Indeed, for the choice xn:¼xŒnŒ in Theorem 9 and using relation(11), we obtain:

Corollary 12. Let (xn) be a sequence of real numbers with xS¼1;d2N^rand d2N^I: We have

Bn;k

d t

i S _p

BjijjSjþt;tðxjÞ jij jSj þt

t

0 BB B@

1 CC CA¼ d

ðk1Þ!T

n nkS;S;. . .;S

p

BjnjkjSjþT;TðxjÞ jnj kjSj þT

T

; ð17Þ

where t =dÆ(iS) + d, T =dÆ(nkS) + kd.

Example 13. Leta,xbe real numbers and (fn(x)) be a binomial type sequence of poly- nomials. Then, forxn¼_anþx^nx fn1ðanþxÞ;we obtain from the identity (see Proposition given in [3])

Bn;k

nx

anþxf_n1ðanþxÞ

¼ n k

kx

aðnkÞ þkxf_nkðaðnkÞ þkxÞ the following identity

Bn;k dx i S _p

f_nkðaðjij jSjÞ þtxÞ aðjij jSjÞ þtx

¼ dx ðk1Þ!

n nkS;S;. . .;S

p

f_nkðaðjnj kjSjÞ þTxÞ aðjnj kjSjÞ þTx ; wheret=dÆ(iS) +d,T=dÆ(nkS) +kd.

Theorem 14. Let (x_n) be a sequence of real numbers with x_S= 1, d be an integer and d2 ðN^IÞ^r. Then, ford.n+ dP1, we have

A_n ðd:i1Þ!d iþ ðd:iÞS i;S;. . .;S 1

p

Biþðd:iÞS;d:i

j S _px_j !

¼ ðd:nþd1Þ!d nþ ðd:nþdÞS n;S;. . .;S 1

p

Bnþðd:nþdÞS;d:nþd

j S _px_j

: ð18Þ

Proof. Seta=d andx=d, Identity(10)becomes An df_iðd:iÞ

d:i

¼df_nðd:nþdÞ

d:nþd : ð19Þ

From (15)we obtain fiðd:iÞ

d:i ¼ðd:i1Þ! iþ ðd:iÞS i;S;. . .;S ¹

p

Biþðd:iÞS;d:i

j S _pxj

f_nðd:nþdÞ

d:nþd ¼ ðd:nþd1Þ! nþ ðd:nþdÞS n;S;. . .;S ¹

p

Bnþðd:nþdÞS;d:nþd

j S _pxj

: Now, to obtain (18), replace in (19) f_i(d.i) andf_n(d.n+d) by their above expressions. h

Corollary 15. Let (xn) be a sequence of real numbers with x_S= 1. Then, for r = 1, S= u andd= pP1, with p, d are integers and pn + dP1, and An(xi) be the complete Bell polynomials and we get

An ðpi1Þ!d ðpuþ1Þi i;u;. . .;u 1

B_{ðpuþ1Þi;pi} j u xj

!

¼ ðpnþd1Þ!d ðpuþdþ1Þn n;u;. . .;u 1

Bðpuþdþ1Þn;pnþd

j u xj

:

Corollary 16. Let (xm,n) be a sequence of real numbers with x_S= 1 and pP1, qP1, d be integers such that pm + qn + dP 1. Then, for r = 2,d= (p,q) andS= (u,0), we get

Am;n ðpiþqj1Þ!d iþ ðpiþqjÞu i;u;. . .;u 1

BiþðpiþqjÞu;j;piþqj

i⁰ u x_i⁰_;j⁰ !

¼ ðpmþqnþd1Þ!d mþ ðpmþqnÞu m;u;. . .;u 1

BmþðpmþqnþdÞu;n;pmþqnþd

i⁰ u x_i⁰_;j⁰

;

for r = 2,d= (p,q) andS= (0,v), we get Am;n ðpiþqj1Þ!d jþ ðpiþqjÞv

j;v;. . .;v 1

Bi;jþðpiþqjÞv;piþqj

j⁰ v xi⁰;j⁰

!

¼ ðpmþqnþd1Þ!d nþ ðpmþqnÞv n;v;. . .;v 1

Bm;nþðpmþqnþdÞv;pmþqnþd

j⁰ v xi⁰;j⁰

; for r = 2,d= (p,q) andS= (u,v), we get

Am;n ðpiþqj1Þ!d

BiþðpiþqjÞu;jþðpiþqjÞv;piþqj

i⁰ u

j⁰ v xi⁰;j⁰

iþ ðpiþqjÞu i;u;. . .;u

jþ ðpiþqjÞv j;v;. . .;v

0 BB B@

1 CC CA

¼ ðpmþqnþd1Þ!d

BmþðpmþqnþdÞu;nþðpmþqnþdÞv;pmþqnþd

i⁰ u

j⁰ v xi⁰;j⁰

mþ ðpmþqnÞu m;u;. . .;u

nþ ðpmþqnÞv

n;v;. . .;v

:

Foru= 1 in Corollary 15 we obtain Proposition 4 of[3]and foru=v= 1 in Cor- ollary 16 we obtain Theorem 13 and 16 of[7].

For the choicexn:¼xŒnŒin Theorem 14 and using relation(11), we obtain:

Corollary 17. Let (xn) be a sequence of real numbers with xS= 1,d be an integer and d2N^r:Then, ford.n + dP1, we have

An

d d:i

Bðdþ1Þ:i;d:iðxjÞ ðdþ1Þ:i

d:i 0

BB B@

1 CC CA¼ d

d:nþd

Bðdþ1Þ:nþd;d:nþdðxjÞ ðdþ1Þ:nþd

d:nþd :

Example 18. Leta,xbe real numbers and (fn(x)) be a binomial type sequence of poly- nomials. For the (xn) given in Example 13, we obtain

An dxfjijðajij þ ðd:iÞxÞ ajij þ ðd:iÞx

¼dxfjnjðajnj þ ðd:nþdÞxÞ ajnj þ ðd:nþdÞx :

4. MULTIPARTITIONAL POLYNOMIALS AND DERIVATIVES OF POLYNOMIALS OF MULTINOMIAL TYPE

Some identities related to the derivatives of polynomial sequences of multinomial type are given. We use the convention D¹_z¼0gðzÞ ¼0:

Theorem 19. Leta2Rbe a real number and (fn(x)) is of multinomial type sequence of polynomials. We have

Bn;k

i

S _pD_z¼0ðe^azf_iSðxþzÞÞ

¼ 1 k!

n nkS;S;. . .;S

p

D^k_z¼0ðe^azf_nkSðkxþzÞÞ: ð20Þ

Proof. Letxn¼ n

S _pD_z¼0ðe^azf_nSðxþzÞÞ. Then

1 k!

P

jijP1xitⁱ i!

k

¼_k!¹ P

iPS

i

S _pD_z¼0ðe^azf_iSðxþzÞÞ^t_i!ⁱ

k

¼ ^tkS

k!ðS!Þ^k

P

iP0D_z¼0ðe^azf_iðxþzÞÞ^t_i!ⁱ k

¼ ^tkS

k!ðS!Þ^k Dz¼0e^azP

iP0fiðxþzÞ^t_i!ⁱ k

¼ ^tkS

k!ðS!Þ^kðDz¼0ðe^azFðtÞ^xþzÞÞ^k

¼ ^tkS

k!ðS!Þ^kðFðtÞ^xDz¼0eðaþlnFðtÞÞzÞ^k

¼ ^tkS

k!ðS!Þ^kFðtÞ^kxðaþlnFðtÞÞ^k

¼ ^tkS

k!ðS!Þ^kFðtÞ^kxD^k_z¼0e^{ðaþlnFðtÞÞz}

¼ ^tkS

k!ðS!Þ^kD^k_z¼0ðe^azFðtÞ^kxþzÞ

¼ ^tkS

k!ðS!Þ^k

P

iP0D^k_z¼0ðe^azfiðkxþzÞÞ^t_i!ⁱ

¼_k!¹P

nPkS

n nkS;S;. . .;S

p

D^k_z¼0ðe^azf_nkSðkxþzÞÞ^t_n!ⁿ;

so, we obtain the desired identity by identification. h

Theorem 20. Let d2N^r, d2N^I,a2R^r,a, x be real numbers and (fn(x)) be a multi- nomial type sequence of polynomials. Then we have

Bn;k d i

S _pD^d:ðiSÞþd1ðxDþ1Þ f_iSðtþzÞ

tþz e^az

z¼0

¼ d ðk1Þ!

n nkS;S;. . .;S

p

Dd:ðnkSÞþkd1ðxDþ1Þ fnkSðTþzÞ ðTþzÞ e^az

z¼0

;

ð21Þ wheret¼a:ðiSÞ þdx and T¼a:ðnkSÞ þkdx.

Proof. Set in(14)x_n:¼Dz=0(e^azf_nS(x+z)) and use Theorem 19, we obtain Bn;k

d t

i

S _pD^t_z¼0ðe^azf_iSðtxþzÞÞ

¼ d

ðk1Þ!T

n nkS;S;. . .;S

p

D^T_z¼0ðe^azf_nkSðTxþzÞÞ; witht=dÆ(iS) +d,T=dÆ(nkS) +kd.

To obtain (21), use in this identity the multinomial type sequence (hn(x)) instead (fn(x)), wherehn(x) is defined, as in(8), byhnðxÞ:¼_ðaxdÞ:nþx^x fnððaxdÞ:nþxÞ. h Corollary 21. Let a, x, a be real numbers, pP0, dP1 be integers and (fn(x)) be a binomial type sequence of polynomials. For r = 1,d= p,a= a andS= u, we have

Bn;k d i

u D^pðiuÞþd1ðxDþ1Þ fiuðt1þzÞ t1þz e^az

z¼0

¼ d ðk1Þ!

n ku

D^{pðnkuÞþkd1}ðxDþ1Þ fnkuðT1þzÞ T1þz e^az

z¼0

;

wheret1¼aðiuÞ þdx and T1¼aðnkuÞ þkdx.

Corollary 22. Let a,x,a, b be real numbers, pP0, qP0, dP1 be integers and (fm,n(x)) be a trinomial type sequence of polynomials. Then, for r = 2, d= (p, q), a= (a, b) andS= (u, 0), we get

Bm;n;k d i

u DpðiuÞþqjþd1ðxDþ1Þ f_iu;jðt10þzÞ t10þz e^az

z¼0

¼ d ðk1Þ!

m ku

DpðmkuÞþqnþkd1ðxDþ1Þ f_mku;nðT10þzÞ T10þz e^az

z¼0

;

where t10¼aðiuÞ þbjþdx and T10¼aðmkuÞ þbnþkdx; for r = 2,d= (p,q), a= (a,b) andS= (0,v), we get

Bm;n;k d j

v DpiþqðjvÞþd1ðxDþ1Þ f_i;jvðt01þzÞ t01þz e^az

z¼0

¼ d ðk1Þ!

n kv

DpmþqðnkvÞþkd1ðxDþ1Þ f_m;nkvðT01þzÞ T01þz e^az

z¼0

;

where t01¼aiþbðjvÞ þdx and T01¼amþbðnkvÞ þkdx; for r = 2, d= (p,q), a= (a,b) andS= (u,v), we get

Bm;n;k d i u

j

v DpðiuÞþqðjvÞþd1ðxDþ1Þ f_iu;jvðt11þzÞ

t11þz e^az

z¼0

¼ d ðk1Þ!

m ku n

kv

DpðmkuÞþqðnkvÞþkd1ðxDþ1Þ f_mku;nkvðT11þzÞ ðT11þzÞ e^az

z¼0

; where t11¼aðiuÞ þbðjvtÞ þdx and T11¼aðmkuÞ þbðnkvÞ þkdx.

Foru= 1 in Corollary 21 we obtain Corollary 19 of[5]and foru=v= 1 in Cor- ollary 22 we obtain Theorems 16 and 17 of [5].

Now, when we use a multinomial type sequence as in (9), Theorem 20 becomes:

Corollary 23. Let d2N^r, d2N^I, 2R^r, a, x be real numbers and (fn(x)) be a multinomial type sequence of polynomials. Then we have

Bn;k d i

S _pD^d:ðiSÞþd1ðxDþ1Þ fjiSjðtþzÞ

tþz e^az

j_z¼0

¼ d ðk1Þ!

n nkS;S;. . .;S

p

Dd:ðnkSÞþkd1ðxDþ1Þ fjnkSjðTþzÞ ðTþzÞ e^az

z¼0

;

ð22Þ where t¼a:ðiSÞ þdx and T¼a:ðnkSÞ þkdx.

Theorem 24. Letd2 ðN^IÞ^r;d be an integer,a2R^r;a;x be real numbers and (f_n(x)) be a multinomial type sequence of polynomials. Then, ford.n + dP1, we have

An dD^d:i1ðxDþ1Þ fiða:iþzÞ a:iþz e^az

z¼0

¼dD^d:nþd1ðxDþ1Þ fnða:nþdxþzÞ a:nþdxþz e^az

z¼0

: ð23Þ

Proof. Set in(18)xn:¼Dz=0(e^azf_nS(x+z)) and use Lemma 19, we obtain An

d

d:iD^d:i_z¼0ðe^azfiððd:iÞxþzÞÞ

¼ d

d:nþdD^d:nþd_z¼0 ðe^azfnððd:nþdÞxþzÞÞ:

To obtain (23), use in this identity the multinomial type sequence (hn(x)) instead of (fn(x)), wherehn(x) is defined, as in(8), byhnðxÞ:¼_ðaxdÞ:nþx^x fnððaxdÞ:nþxÞ. h

Corollary 25. Leta, x, a be real numbers, pP1, d be integers such that pn + dP1 and (fn(x)) be a binomial type sequence of polynomials. For r = 1,d= p anda= a, we get

An dD^pi1ðxDþ1Þ fiðaiþzÞ aiþz e^az

z¼0

¼dD^pnþd1ðxDþ1Þ fnðanþdxþzÞ

anþdxþz e^az

z¼0

:

Corollary 26. Let a, x, a, b be real numbers, pP1, qP1, d be integers such that pm + qn + dP1 and (fm,n(x)) be a trinomial type sequence of polynomials. For r = 2,d= (p,q) and a= (a,b), we get

Am;n dD^piþqj1ðxDþ1Þ fi;jðaiþbjþzÞ aiþbjþz e^az

z¼0

¼dD^pmþqnþd1ðxDþ1Þ fm;nðamþbnþdxþzÞ amþbnþdxþz e^az

z¼0

:

Corollary 25 is Corollary 20 of[5]and Corollary 26 is Theorem 18 of[5].

Now, when we use a multinomial type sequence as in(9), Theorem 24 becomes:

Corollary 27. Letd2 ðN^IÞ^r;d be an integer,a2R^r;a, x be real numbers and (f_n(x)) be a multinomial type sequence of polynomials. Then, ford.n + dP1, we have

An dD^d:i1ðxDþ1Þ fjijða:iþzÞ a:iþz e^az

z¼0

¼dD^d:nþd1ðxDþ1Þ f_jnjða:nþdxþzÞ a:nþdxþz e^az

z¼0

:

ACKNOWLEDGMENTS

The authors thank the referee for valuable advice and comments which helped to im- prove the quality of this paper.

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