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Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 13 (2012), No 2, pp. 197-208 DOI: 10.18514/MMN.2012.356

The matrix version for the multivariable

Humbert polynomials

Rabia Akta³, Bayram Çekim, and Recep �ahin

Miskolc Mathematical Notes HU e-ISSN 1787-2413Vol. 13 (2012), No. 2, pp. 197–208

THE MATRIX VERSION FOR THE MULTIVARIABLE HUMBERTPOLYNOMIALS

RABIA AKTAS, BAYRAM CEKIM, AND RECEP SAHIN

Received 4 May, 2011

Abstract. In this paper, the matrix extension of the multivariable Humbert polynomials is int-roduced. Various families of linear, multilinear and multilateral generating matrix functions ofthese matrix polynomials are presented. Miscellaneous applications are also discussed.

2000 Mathematics Subject Classification: 33C25; 15A60

Keywords: Humbert polynomials, Chan-Chyan-Srivastava polynomials, Lagrange-Hermite poly-nomials, generating matrix function, matrix functional calculus

1. INTRODUCTION

It is well-known that special matrix functions appear in the study of many areas.Generalization of the property of orthogonality [11, 12], Rodrigues formula [6, 10],a second-order Sturm-Liouville differential equation [10], a three-term matrix re-currence [6, 7], relation between different orthogonal matrix polynomials [21] aretheoretical examples. Statistics, group representation theory [17], scattering theory[15], differential equations [18, 19], Fourier series expansions [9], interpolation andquadrature [22, 23], splines [8], and medical imaging [5] are areas of application oforthogonal matrix polynomials.

Throughout this paper, for a matrix A 2CN�N , its spectrum is denoted by �.A/.The two-norm of A; which will be denoted by kAk ; is defined by

kAk D supx¤0

kAxk2kxk2

;

where, for a vector y 2CN , kyk2 D�yT y

�1=2is the Euclidean norm of y: I and �

will denote the identity matrix and the null matrix in CN�N , respectively. We saythat a matrix A in CN�N is a positive stable if <.�/ > 0 for all � 2 �.A/, where�.A/ is the set of the eigenvalues of A. If A0;A1; :::;An are elements of CN�N andAn ¤ � , then we call

P.x/D AnxnCAn�1x

n�1C :::CA1xCA0

c 2012 Miskolc University Press

198 RABIA AKTAS, BAYRAM CEKIM, AND RECEP SAHIN

a matrix polynomial of degree n in x. From [20], one can see

.P /n D P.P CI /.P C2I /:::.P C .n�1/I /I n� 1I .P /0 D I: (1.1)

For any matrix A in CN�N , the authors exploited the following relation due to [20]

.1�x/�A D

1XnD0

.A/n

nŠxn ; jxj< 1: (1.2)

Gould [16] presented a systematic study of an interesting generalization of the Hum-bert and the Gegenbauer polynomials and several other polynomial systems, that iscalled generalized Humbert polynomials and defined by

�C �mxtCytm

�pD

1XnD0

Pn .m;x;y;p;C /tn (1.3)

wherem is a positive integer and the other parameters are unrestricted (see also [26, p.77, 86] ).

Aktas et al. [1] present a systematic investigation of a multivariable extension ofthe Humbert polynomials generated by

rQiD1

˚.Ci �mixi tCyi t

mi /�˛iD

1PnD0

P.˛1;:::;˛r /n .m;x;y;C/tn

.jmixi t �yi tmi j< jCi j I i D 1;2; :::; r/

(1.4)

where xD.x1; :::;xr/ ;yD.y1; :::;yr/ ; CD.C1; :::;Cr/ ; mD.m1; :::;mr/, mi D1;2; ::: .i D 1;2; :::; r/ and the other parameters are unrestricted.

The main objective of this paper is to construct a matrix version of the multiva-riable Humbert polynomials given by (1.4) and the derivation of various familiesof multilinear and mixed multilateral generating matrix functions for these matrixpolynomials. We present some special cases of our results and also obtain severalrecurrence relations for these matrix polynomials.

2. MATRIX EXTENSION OF THE MULTIVARIABLE HUMBERT POLYNOMIALS

The main object of this section is to present a systematic investigation of the matrixextension of the multivariable Humbert polynomials generated by

rQiD1

n.Ci �mixi tCyi t

mi /�Ai

oD

1PnD0

P.A1;:::;Ar /n .m;x;y;C/tn

.jmixi t �yi tmi j< jCi j I i D 1;2; :::; r/

(2.1)

where Ai 2CN�N ;xD.x1; :::;xr/ ;yD.y1; :::;yr/ ; CD.C1; :::;Cr/ ;mD.m1; :::;mr/ ; mi D 1;2; ::: .i D 1;2; :::; r/ and the other parameters are unrestricted:

THE MATRIX VERSION FOR THE MULTIVARIABLE HUMBERT POLYNOMIALS 199

(2.1) yields the following explicit representation:

P .A1;:::;Ar /n .m;x;y;C/

D

Xm1k1C:::CmrkrCn1C:::CnrDn

.A1/n1Ck1C�A1�.n1Ck1/I1 ::: .Ar/nrCkr

C�Ar�.nrCkr /Ir

n1Š:::nr Šk1Š:::kr Š

�mn1

1 :::mnrr .�1/

k1C:::Ckr xn1

1 :::xnrr y

k1

1 :::ykrr

D

Xm1k1C:::CmrkrCn1C:::CnrDn

rYpD18<:

�Ap�npCkp

C�Ap�.npCkp/Ip

npŠkpŠmnp

p .�1/kp xnp

p ykp

p

9=; (2.2)

where, as usual, .A/n denotes the Pochhammer symbol given by (1.1).We notice that the case r D 1 in (2.1) reduces to the matrix version of the gene-

ralized Humbert polynomials introduced by Gould [16]. In this case, it is generatedby �

C �mxtCytm��AD

1XnD0

P .A/n .m;x;y;C /tn (2.3)

where jmxt �ytmj < jC j ; A 2 CN�N ; m is a positive integer and the other para-meters are unrestricted. For the special cases of (2.3), including Gegenbauer matrixpolynomials, we refer [19].

It is clear that the case

Ci D 1; mi D 1; yi D 0 ; i D 1;2; :::; r

of the polynomials of (2.1) reduces to matrix version of the Chan-Chyan-Srivastavamultivariable polynomials, which is generated by [14]

rQiD1

n.1�xi t /

�Ai

oD

1PnD0

g.A1;:::;Ar /n .x1; :::;xr/ t

n�Ai 2CN�N .i D 1;2; :::; r/ I jt j<min

njx1j�1 ; :::; jxr j

�1o�:

(2.4)

Since Ai D ˛i 2 C for N D 1 in (2.4), we obtained the generating function of theChan-Chyan-Srivastava multivariable polynomials [3].

On the other hand, if we choose Ci D 1; mi D i; xi D 0; yi D �xi ; i D

1;2; :::; r in (2.1), we get a matrix version of the multivariable Lagrange-Hermite


polynomials, which is generated by [14]

rQiD1

n�1�xi t

i��Ai

oD

1PnD0

h.A1;:::;Ar /n .x1; :::;xr/ t

n

�Ai 2CN�N .i D 1;2; :::; r/ I jt j<min

njx1j�1 ; jx2j

�1=2 ; :::; jxr j�1=r

o�:

(2.5)Since Ai D ˛i 2C for N D 1 in (2.5), we have the multivariable Lagrange-Hermitepolynomials presented by Altın and Erkus [2]. Furthermore, we should remarkthat the case r D 2 of the polynomials corresponds to the familiar (two-variable)Lagrange-Hermite polynomials considered by Dattoli et al. [4].

Moreover, the special case

Ci D 1; xi D 0; yi D�xi ; i D 1;2; :::; r

gives the matrix version of the Erkus-Srivastava multivariable polynomials generatedby [14]

rQiD1

n.1�xi t

mi /�Ai

oD

1PnD0

u.A1;:::;Ar /n .x1; :::;xr/ t

n;

Ai 2CN�N .i D 1;2; :::; r/ ;

jt j<minnjx1j�1=m1 ; jx2j

�1=m2 ; :::; jxr j�1=mr

o:

(2.6)

Since Ai D ˛i 2 C for N D 1 in (2.6), we have the Erkus-Srivastava multivariablepolynomials generated by [13].

3. AN APPLICATION OF SRIVASTAVA’S THEOREM ON MIXED GENERATINGFUNCTIONS

Srivastava [25] (see also the subsequent treatise on the subject by Srivastava andManocha [26, p. 378, Theorem 12]) obtained a family of mixed generating functionsfor certain general multivariable and multiparameter sequences of functions. Ourgenerating function (2.1) fits easily into the general setting of Srivastava’s theorem.Thus, by applying this general result to the generating function (2.1), we obtain the


following family of mixed generating functions for matrix version of the multivaria-ble Humbert polynomials given by (2.1)

1XnD0

P .A1C�1nI;:::;ArC�rnI/n .m;x;y;C/tn D

rQiD1

n.Ci �mixi�Cyi�

mi /�Ai

o1� �

�rPiD1

�imi .Ci Cyi�mi /�1��yi�mi�1Cxi

.Ci �miyi�mi Cyi�

mi /

.Ci �mixi�Cyi�mi /

�� ;(3.1)

� D � .t/ WD t

rYiD1

n�Ci �mixi�Cyi�

mi��i

oI �i 2C;

Ci ¤ 0 and Ai 2CN�N .i D 1; :::; r/�

where all of the matrices commute with each other. In a special case, it is easilyseen that (3.1) would at once reduce to the generating function (2.1) when �i D0 .i D 1; :::; r/ : For the special case of N D 1; (3.1) gives mixed generating functionfor the multivariable Humbert polynomials given by [1]. Furthermore, the specialcase of N D 1 and r D 1 of (3.1) reduces to the mixed generating function for thegeneralized Humbert polynomials in [24].

4. BILINEAR AND BILATERAL GENERATING MATRIX FUNCTIONS

In this section, we derive several families of bilinear and bilateral generating mat-rix functions for matrix version of the multivariable Humbert polynomials which aregenerated by (2.1) and given explicitly by (2.2).

We begin by stating the following theorem.

Theorem 1. Corresponding to an identically non-vanishing function ˝�.z/ of scomplex variables ´1; :::;´s .s 2N/ and of complex order �, let

��;�.zIw/ WD1XkD0

ak˝�C�k.z/wk (4.1)

where .ak ¤ 0; �;� 2C/ ; zD.´1; :::;´s/ and

�n;p;�;�.x;yIzI�/ WDŒn=p�XkD0

akP.A1;:::;Ar /

n�pk.m;x;y;C/ ˝�C�k.z/�k (4.2)


where n;p 2 N; Ai 2 CN�N I xD.x1; :::;xr/ I yD.y1; :::;yr/ ICD.C1; :::;Cr/ ImD.m1; :::;mr/ ; mi D 1;2; ::: .i D 1;2; :::; r/: Then we have1XnD0

�n;p;�;�

�x;yIzI

�

tp

�tn D

rYiD1

n�Ci �mixi tCyi t

mi��Ai

o��;�.zI�/ (4.3)

provided that each member of (4.3) exists.

Proof. Let T denote the left-hand side of the equality (4.3) of Theorem 1. Then,upon substituting the polynomials

�n;p;�;�

�x;yIzI

�

tp

�from definition (4.2) into the left-hand side of (4.3), we find

T D

1XnD0

Œn=p�XkD0

akP.A1;:::;Ar /

n�pk.m;x;y;C/˝�C�k.z/�ktn�pk : (4.4)

Replacing n by nCpk; we can write

T D

1XnD0

1XkD0

akP.A1;:::;Ar /n .m;x;y;C/˝�C�k.z/�ktn

D

1XnD0

P .A1;:::;Ar /n .m;x;y;C/tn

1XkD0

ak˝�C�k.z/�k

D

rYiD1


mi��Ai

o��;�.zI�/;

which completes the proof. �

In a similar manner, we can give the next result.

Theorem 2. For a non-vanishing function ˝�.z/ of s complex variables´1; : : :´s .s 2N/ and for p 2N, �;� 2C, zD.´1; :::;´s/ ; A WD .A1; :::;Ar/ ;B WD .B1; :::;Br/ ; Ai ;Bi 2CN�N for i D 1;2; :::; r; let

�n;p�;�;C;m.x;yIzIw/ WD

Œn=p�XkD0

akP.A1CB1;:::;ArCBr /

n�pk.m;x;y;C/˝�C�k.z/wk (4.5)

where ak ¤ 0I n;k 2N0; N0 WDN[f0g. Then we have

nXkD0

Œk=p�XlD0

alP.A1;:::;Ar /

n�k.m;x;y;C/P .B1;:::;Br /

k�pl.m;x;y;C/ ˝�C�l.z/wl

D�n;p�;�;C;m.x;yIzIw/ (4.6)


provided that each member of (4.6) exists where the matrices commute with eachother.

5. SPECIAL CASES AND SOME FURTHER PROPERTIES

It is possible to give many applications of the theorems obtained in the previoussections with the help of appropriate choices of the multivariable functions˝�C�k.z/ ;zD.´1; :::;´s/ ; k 2N0 ; s 2N. For example, if we set

s D r and ˝�C�k.z/D h.B1;:::;Br /

�C�k.z/

in Theorem 1, where the matrix version of the multivariable Lagrange-Hermite poly-nomials

h.B1;:::;Br /n .x/

are generated by (2.5), then we obtain the following result which provides a classof bilateral generating matrix functions for the matrix version of the multivariableLagrange-Hermite polynomials and for the matrix version of the multivariable Hum-bert polynomials given explicitly by (2.2).

Corollary 1. If ��;�.zIw/ WD1PkD0

akh.B1;:::;Br /

�C�k.z/wk ;ak ¤ 0; �;� 2N0;

zD .´1; :::;´r/ and

�n;p;�;�.x;yIzI�/ WDŒn=p�XkD0

akP.A1;:::;Ar /

n�pk.m;x;y;C/h.B1;:::;Br /

�C�k.z/�k

where n 2N0I p 2N; Ai ;Bi 2CN�N ; xD.x1; :::;xr/ I yD.y1; :::;yr/ ICD.C1; :::;Cr/ ImD.m1; :::;mr/ ; mi D 1;2; ::: .i D 1;2; :::; r/, then1XnD0

�n;p;�;�

�x;yIzI

�

tp

�tn D

rYiD1


mi��Ai

o��;�.zI�/ (5.1)


Remark 1. Using the generating relation (2.5) for the matrix version of the mult-ivariable Lagrange-Hermite polynomials and setting ak D 1; �D 0; �D 1;we obtain

1XnD0

Œn=p�XkD0

P.A1;:::;Ar /

n�pk.m;x;y;C/h.B1;:::;Br /

k.z/�ktn�pk

D

rYiD1

�Ci �mixi tCyi t

mi��Ai

! rYiD1

.1�´i�i /�Bi

!;

wherej�j<min

nj´1j�1 ; j´2j

�1=2 ; :::; j´r j�1=r

o;


jmixi t �yi tmi j< jCi j I i D 1;2; :::; r:

Also, if we choose sD 2r and˝�C�k.z/DP.E1;:::;Er /

�C�k.m; t;!;C/; �;� 2N0; tD

.t1; :::; tr/ ; ! D .!1; :::;!r/ in Theorem 2, we obtain the following class of bilineargenerating matrix functions for the matrix version of the multivariable Humbert poly-nomials given explicitly by (2.2).

Corollary 2. If

�n;p�;�;C;m.x;yI t;!Iw/

WD

Œn=p�XkD0

akP.A1CB1;:::;ArCBr /

n�pk.m;x;y;C/P .E1;:::;Er /

�C�k.m; t;!;C/wk

.ak ¤0I p 2NIn;k;�;� 2N0/

where Ai ;Bi ;Ei 2CN�N for i D 1;2; :::; r; then

nXkD0

Œk=p�XlD0

alP.A1;:::;Ar /

n�k.m;x;y;C/

P.B1;:::;Br /

k�pl.m;x;y;C/P .E1;:::;Er /

�C�l.m; t;!;C/wl

D�n;p�;�;C;m.x;yI t;!Iw/ (5.2)

provided that each member of (5.2) exists where AiBj D BjAi for i;j D 1;2; :::; r:

For example, if we set

s D 1 and ˝�C�k.y /D L.E;�/

�C�k.y/

in Theorem 1, where the nth Laguerre matrix polynomials L.E;�/n .x/ are defined by[18]

L.E;�/n .x/D

nXkD0

.�1/k �k

kŠ.n�k/Š.ECI /n Œ.ECI /k�

�1xk;

where E is a matrix in CN�N , ECnI is invertible for every integer n � 0 and � isa complex number with <.�/ > 0 and they are generated by

1XnD0

L.E;�/n .x/tn D .1� t /�.ECI/ exp��xt

1� t

�; (5.3)

jt j< 1; 0 < x <1;

then we obtain the following result which provides a class of bilateral generatingmatrix functions for the matrix version of the multivariable Humbert and Laguerrematrix polynomials.


Corollary 3. If ��;�.Íw/ WD1PkD0

akL.E;�/

�C�k.´/wk where .ak ¤ 0; �;� 2N0/I

and

�n;p;�;�.x;yIÍ�/ WDŒn=p�XkD0

akP.A1;:::;Ar /

n�pk.m;x;y;C/L.E;�/

�C�k.´/�k

where n;p 2N: Then we have1XnD0

�n;p;�;�

�xIyIÍ

�

tp

�tn D

rYiD1


mi��Ai

o��;�.Í�/ (5.4)


Remark 2. Using the generating relation (5.3) for the Laguerre matrix polynomialsand taking ak D 1; �D 0; � D 1; we have

1XnD0

Œn=p�XkD0

P.A1;:::;Ar /

n�pk.m;x;y;C/L.E;�/

k.´/�ktn�pk

D

rYiD1


mi��Ai

o� .1��/�.ECI/ exp

��´�

1��

�; (5.5)

where j�j< 1; 0 < ´ <1:

Remark 3. For r D 1 in (5.5), we have a bilateral generating matrix function of theHumbert (2.3) and Laguerre matrix polynomials:

1XnD0

Œn=p�XkD0

P.A/

n�pk.m;x;y;C /L

.E;�/

k.´/�ktn�pk

D�C �mxtCytm

��A.1��/�.ECI/ exp

��´�

1��

�:

Remark 4. For r D 1 and s D 2 in Theorem 1, setting˝�C�k.´/ D P

.B/n .m;x;y;C /.B 2 CN�N / and taking ak D 1; � D 0; � D 1;we

have bilinear generating matrix function for the Humbert matrix polynomials:

1XnD0

Œn=p�XkD0

P.A/

n�pk.m;x;y;C /P

.B/

k.m;x;y;C /�ktn�pk

D�C �mxtCytm

��A �C �mx�Cy�m

��B:


Furthermore, for every suitable choice of the coefficients ak .k 2N0/; if the mult-ivariable function˝�C�k.y/; yD.y1; :::;ys/; .s 2N/; is expressed as an appropriateproduct of several simpler functions, then the assertions of Theorems 1 and 2 can beapplied in order to derive various families of multilinear and multilateral generatingmatrix functions for the matrix version of the multivariable Humbert polynomialsgiven explicitly by (2.2).

We now discuss some further properties of matrix version of the multivariableHumbert polynomials given by (2.2). First of all, the generating matrix relation (2.1)yields the following addition formula for these multivariable polynomials:

P .A1CB1;:::;ArCBr /n .m;x;y;C/

D

nXkD0

P.A1;:::;Ar /

n�k.m;x;y;C/P .B1;:::;Br /

k.m;x;y;C/

where Ai ;Bi 2CN�N ; AiBj D BjAi for i;j D 1;2; :::; r:On the other hand, the multivariable Humbert matrix polynomials satisfy the fol-

lowing equation:rX

jD1

�xj

@

@xjCmjyj

@

@yj

�P .A1;:::;Ar /n .m;x;y;C/D nP .A1;:::;Ar /

n .m;x;y;C/:

(5.6)If we differentiate each member of the generating function (2.1) with respect to xjand yj .j D 1;2; :::; r/; we obtain the following (differential) recurrence relations forthe matrix version of the multivariable Humbert polynomials:

@

@xjP .A1;:::;Ar /n .m;x;y;C/

D

n�1XkD0

Œk=mj �XlD0

.�1/l�kC l � lmj

�ŠAj

�mj�k�lmjC1�

k� lmj�ŠlŠC

k�l.mj�1/C1j

xk�lmj

j ylj

�P.A1;:::;Ar /

n�k�1.m;x;y;C/ (5.7)

for n� 1; and

@

@yjP .A1;:::;Ar /n .m;x;y;C/

D�

n�mjXkD0

Œk=mj �XlD0


�ŠAj

�mj�k�lmj�

k� lmj�ŠlŠC

k�l.mj�1/C1j

xk�lmj

j ylj

�P.A1;:::;Ar /

n�k�mj.m;x;y;C/ (5.8)


where n �mj and mj .j D 1;2; :::; r/ is a positive integer and all matrices are com-mutative. By applying (5.6), (5.7) and (5.8), the following recurrence relation for thematrix polynomials (given explicitly by (2.2)) can be easily derived:

rXjD1

n�1XkD0

Œk=mj �XlD0


�ŠAj


k� lmj�lŠC

k�l.mj�1/C1j

xk�lmjC1

j yljP.A1;:::;Ar /

n�k�1.m;x;y;C/

�

rXjD1

n�mjXkD0

Œk=mj �XlD0


�ŠAj


k� lmj�ŠlŠC

k�l.mj�1/C1j

xk�lmj

j ylC1j P.A1;:::;Ar /

n�k�mj.m;x;y;C/

DnP .A1;:::;Ar /n .m;x;y;C/, for n�mj

where mj .j D 1;2; :::; r/ is a positive integer and all matrices commute.

ACKNOWLEDGEMENT

The authors are grateful to the referee(s) for their valuable comments and suggest-ions which improved the quality and the clarity of the paper.

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Authors’ addresses

Rabia AktasAnkara University, Faculty of Science, Department of Mathematics, Tandogan TR-06100, Ankara,

TurkeyE-mail address: [email protected]

Bayram CekimGazi University, Faculty of Science, Department of Mathematics, Teknik Okullar TR-06500, An-

kara, TurkeyE-mail address: [email protected]

Recep SahinAnkara University, Faculty of Science, Department of Mathematics, Tandogan TR-06100, Ankara,

TurkeyE-mail address: [email protected]

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