Journal of the Association of Arab Universities for Basic and Applied Sciences (2017) 24, 271–276
University of Bahrain
Journal of the Association of Arab Universities for Basic and Applied Sciences www.elsevier.com/locate/jaaubas www.sciencedirect.com
A note on two variable Laguerre matrix polynomials Maged G. Bin-Saad Department of Mathematics, Faculty of Science, Aden University, Aden, Yemen Received 22 November 2015; revised 31 July 2016; accepted 21 September 2016 Available online 14 November 2016
KEYWORDS Laguerre matrix polynomials; Operational images; Functional relations; Differential matrix equations
Abstract The principal object of this paper is to present a natural further step toward the mathematical properties and presentations concerning the two variable Laguerre matrix polynomials defined in (Bin-Saad, Maged G., Antar, A. Al-Sayaad, 2015. Study of two variable Laguerre polynomials via symbolic operational images. Asian J. of math. and comput. research, 2(1), 42–50). Series expansions, integral transforms and bilinear and bilateral generating matrix functions for these polynomials are established. Some particular cases and consequences of our main results are also considered. Ó 2016 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
1. Introduction The subject of Laguerre polynomials has gained importance during the last two decades mainly due to its applications in various fields of mathematical physics, such as the solving of delay differential equations (Suayip et al., 2014), pantographtype Volterra integro-differential equations (Suayip, 2014) and fractional differential equations (Bhrawy and Alghamdi, 2012; Bhrawy et al., 2014a,b; Bhrawy et al., 2015b,c). Numerous other works dealing also with the use of Laguerre polynomials and matrices include those by Ahmadian et al. (2015) in the theory of Tau method for numerical solution of fuzzy fractional kinetic model, by Bhrawy and Taha (2012) and Bhrawy et al. (2015a) in the theory of operational matrix of fractional integration of Laguerre polynomials and generalized Laguerr e–Gauss–Radau schema for first order hyperbolic equation on semi-finite domain (Abdelkawy and Taha, 2012) and so on (see also Bin-Saad and Antar (2015)). Further, matrix E-mail address:
[email protected] Peer review under responsibility of University of Bahrain.
polynomials seen in the study of many area such as statistics, Lie group theory and number theory are well known. Recently, the matrix versions of the classical families orthogonal polynomials such as Laguerre, Jacobi, Hermite, Gegenbauer, Bessel and Humbert polynomials and some other polynomials were introduced by many authors for matrices in CNN and various properties satisfied by them were given from the scalar case, see for example (Aktas et al., 2013; Aktas et al., 2011; Altin and Cekim, 2012a,b, 2013; Bin-Saad and Antar, 2015; Cekim and Erkus-Duman, 2014; Jo´dar and Corte´s, 1998a,b; Jo´dar and Company, 1996; Jo´dar et al., 1995; Pathan et al., 2014; Bayram and Altin, 2015). If A0 ; A1 ; . . . ; An . . . , are elements of CNN and An – 0, then we call PðxÞ ¼ An xn þ An1 xn1 þ An2 xn2 þ þ A1 x þ A0 ; a matrix polynomial of degree n in x. If A þ nI is invertible for every integer n P 0 then. ðAÞn ¼ AðA þ IÞðA þ 2IÞ ðA þ ðn 1ÞIÞ; n P 1; ðAÞ0 ¼ I:
http://dx.doi.org/10.1016/j.jaubas.2016.09.001 1815-3852 Ó 2016 University of Bahrain. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
ð1:1Þ
272
M.G. Bin-Saad
In Bin-Saad and Antar (2015), it is shown that an appropriate combination of methods, relevant to operational calculus and to matrix polynomials, can be a very useful tool to establish and treat a new class of two variable Laguerre matrix polynomials in the following form ðA;kÞ Ln;m ðx; yÞ ¼
n X m X ð1Þsþk ðkxÞs ðkyÞk ðA þ IÞnþm s!k!ðn sÞ!ðm kÞ! s¼0 k¼0
½ðA þ IÞsþk 1 ; fn; mg P 0;
ð1:2Þ
where A be a matrix in CNN where ðaÞ is not an eigenvalue of A for every integer a > 0 and k be a complex number whose real part is positive. The authors in Bin-Saad and Antar (2015) explored the formal properties of the operational identities to derive a number of properties of the new class two variable Laguerre matrix polynomials (1.2) and discussed the links with various known polynomials. The generating relation for the ðA;kÞ matrix function Ln;m ðx; yÞ, is given by the following formula (Bin-Saad and Antar, 2015): kðxu þ yvÞ ½ð1 u vÞðAþIÞ exp 1uv 1 X n m ¼ LðA;kÞ ð1:3Þ n;m ðx; yÞu v ; where fu; v; x; yg 2 C and ju þ vj < 1. By setting m ¼ y ¼ 0 in (1.2), Eq. (1.2) immediately yields the following Laguerre matrix polynomials due to Jo´dar and Corte´s (1998a,b):
Theorem 2.1. Let A and B be matrices in CNN satisfying spectral condition ðaÞ is not an eigenvalue of A for every integer a > 0; ðbÞ is not an eigenvalue of B for every integer b > 0; AB ¼ BA; RðkÞ > 0 and n P 0; m P 0. Then n X m X ðA þ BÞnþmsk ðB;kÞ Ls;k ðx; yÞ ¼ LðA;kÞ n;m ðx; yÞ: ðn sÞ!ðm kÞ! s¼0 k¼0
ð2:1Þ
ð1:4Þ For the purpose of this work, we recall here same definitions. Definition 1.1. Let A be a positive stable matrix in CNN , then Gamma matrix function is defined by (Jo´dar and Corte´s, 1998a,b) Z
1
CðAÞ ¼
e1 tAI dt:
ð1:5Þ
0
Definition 1.2. Let A, B and A þ B be a positive stable matrix in CNN and AB ¼ BA, then Beta matrix function is defined by (Jo´dar and Corte´s, 1998a,b) bðA; BÞ ¼ CðAÞC1 ðBÞCðA þ BÞ ¼
Z
1
tAI ð1 tÞBI dt:
ð1:6Þ
0
Lemma 1.1. For matrix Aðk; nÞ in CNN where n P 0; k P 0 , we have (see Srivastava and Manocha (1984)): 1 X 1 1 X n X X Aðn; kÞ ¼ Aðn; n kÞ;
ð1:7Þ
and
n¼0 k¼0
m X ðA þ BÞnþm ðB;kÞ Ls;k ðx; yÞunþs vmþk ; n!m! n;m;s;k¼0
which on using the multinomial formula ð1 x yÞk ¼
1 X ðkÞnþm xn ym ; n!m! n;m¼0
gives us 1 X n X m X ðA þ BÞnþmsk ðB;kÞ Ls;k ðx; yÞun vm ðn sÞ!ðm kÞ! n;m¼0 s¼0 k¼0
¼ ð1 u vÞðABÞ
1 X
ð1:8Þ
ðB;kÞ
Ls;k ðx; yÞus vk :
ð2:2Þ
s;k¼0
Now, employing (1.3) in (2.2) and comparing the coefficients of un vm in the resulting expression, we get the desired result. h Theorem 2.2. Let A be matrix in CNN satisfying spectral condition ðaÞ is not an eigenvalue of A for every integer a > 0; RðkÞ > 0; fjzj; jwjg < 1 n P 0; m P 0. Then n X m X ðA þ ðs þ k þ 1ÞIÞ
n¼0 k¼0
1 X n 1 X 1 X X Aðn; kÞ ¼ Aðn; n þ kÞ:
1 X n X m X ðA þ BÞnþmsk ðB;kÞ Ls;k ðx; yÞun vm ðn sÞ!ðm kÞ! n;m¼0 s¼0 k¼0
¼
n X ð1Þs LnðA;kÞ ðxÞ ¼ ðA þ IÞn ½ðA þ IÞs 1 ðkxÞs ; k P 0: s!ðn sÞ! s¼0
n¼0 k¼0
2. Finite and infinite sums
Proof. Using (1.8), we can write
n;m¼0
n¼0 k¼0
Motivated by the important role of the Laguerre matrix polynomials in several diverse fields of physics and the contributions in Bayram and Altin (2013) and Jo´dar et al. (1994) toward the generalization of the Laguerre polynomials, this work aims at investigating several properties for the two variable Laguerre matrix polynomials LðA;kÞ n;m ðx; yÞ. We establish some projection series, integral transforms and bilinear and bilateral generating matrix functions. Many earlier (known) results given by Bayram and Altin (2013) are shown to be special cases of our results.
s¼0 k¼0
zÞn ð1 wÞm ðA;kÞ Ls;k ðx; yÞ ðn sÞ!ðm kÞ! nþmsk ð1
z s w k ¼ LðA;kÞ n;m ðxz; ywÞ; 1z 1w
Proof. According to formula (1.8), we can write
ð2:3Þ
A note on two variable Laguerre matrix polynomials 1 X n X m X ðA þ ðs þ k þ 1ÞIÞ
nþmsk ð1 zÞ
n
273 1 X zs wk
ð1 wÞm
ðn sÞ!ðm kÞ!
n;m¼0 s¼0 k¼0
s;k¼0
1 z s w k X ðA;kÞ ðA;kÞ un vm ¼ Ls;k ðx; yÞ Ls;k ðx;yÞ 1z 1w s;k¼0 ! 1 X ðA þ ðs þ k þ 1ÞIÞnþm ½ð1 zÞun ½ð1 wÞvm ðzuÞs ðwvÞk n!ðm!Þ s;k¼0
¼ ð1 XÞðAþIÞ
1 X
ðA;kÞ
Ls;k ðx;yÞ
s;k¼0
zu ð1 XÞ
s
wv ð1 XÞ
k ;
1 X n X m X ðA þ ðs þ k þ 1ÞIÞnþmsk ð1 zÞn ð1 wÞm ðn sÞ!ðm kÞ! n;m¼0 s¼0 k¼0 z s w k ðA;kÞ un vm Ls;k ðx; yÞ 1z 1w kðxzu þ ywvÞ ¼ ð1 u vÞðAþIÞ exp 1uv 1 X n m ¼ LA;k n;m ðxz; ywÞu v :
ðx; yÞun vm LðAþðsþkÞI;kÞ n;m
¼ ezþw LðA;kÞ n;m ðkx z w; ky z wÞ;
ð2:6Þ
Proof. We have 1 1 X X zs wk ðAþðsþkÞI;kÞ L ðx; yÞun vm s!k! n;m n;m¼0 s;k¼0
¼
where X ¼ ðu þ v zu wvÞ, which in view of (1.3), yields the result
s!k!
1 X
1 X
s;k¼0
n;m¼0
LðAþðsþkÞI;kÞ ðx; yÞun vm n;m
!
zs wk s!k!
1 kðxu uvÞ X zs wk ð1 u vÞðAþðsþkþ1ÞI ¼ exp 1 u v s;k¼0 s!k! uðkx z wÞ vðky z wÞ ¼ ezþw ð1 u vÞAI exp 1uv 1 X kx z w ky z w n m ; uv : ¼ ezþw LðA;kÞ n;m k k n;m¼0 ð2:7Þ
ð2:4Þ
Now, comparing the coefficients of un vm in (2.7), we get the desired result. h
Now, comparing the coefficients of un vm in (2.4), we get the desired result. h
It is important to note that Eqs. (2.1), (2.3), (2.5) and (2.6) provide a generalization of known results in Bayram and Altin (2013) and Eqs. (2.1), (2.4), (2.6) and (2.7).
Theorem 2.3. Let A and C be matrices in CNN satisfying spectral condition ðaÞ is not an eigenvalue of A for every integer a > 0; AC ¼ CA; ju þ vj < 1 and RðkÞ > 0. Then
3. Integral transforms
n;m¼0
1 X
n m ðCÞnþm ½ðA þ IÞnþm 1 LðA;kÞ n;m ðx; yÞu v
n;m¼0
kðxu þ yvÞ ; ¼ ð1 u vÞC F C; A þ I; 1 1 1uv
ð2:5Þ
Theorem 3.1. Let A be matrix in CNN satisfying spectral condition ðaÞ is not an eigenvalue of A for every integer a > 0 and RðkÞ > 0 . Then Z 1 ðA þ IÞnþm C1 ðBÞ ðA;kÞ F1 ek kBI Ln;m ðx; yÞdk ¼ n!m! 0 ½A þ I; nI; mI; B; x; y; ð3:1Þ
Proof. In view of the identity 1 X
fðn þ mÞ
n;m¼0
In many situations an integral transform of Laguerre function is more convenient to use than its series representation. First of all, we establish an integral transform for LðA;kÞ n;m ðx; yÞ involving double series.
1 xn ym X ðx þ yÞn ¼ fðnÞ n! m! n¼0 n!
we can write kðxu þ yvÞ ð1 u vÞC 1 F1 C; A þ I; 1uv 1 1 X ðCÞ ½ðA þ IÞ sþk sþK ðkxuÞs ðkyvÞk ð1 u vÞCðsþkÞI ; ¼ s!k! s;k¼0
where F1 is Applls´ function (Srivastava and Karlsson, 1985). Proof. Denote, for convenience, the left-hand side of Eq. (3.1) by I,then I¼
n X m X s¼0
Z
which on employing the Taylor expression ð1 u vÞCðsþkÞI ¼
ek kBþðsþk1ÞI dk
0
¼
n X m X s¼0
h
Theorem 2.4. Let A be matrix in C satisfying spectral condition ðaÞ is not an eigenvalue of A for every integer a > 0 and RðkÞ > 0 . Then NN
1
1 X ðCÞnþmþsþk ½ðCÞsþk 1 un vm n!m! n;m¼0
and then using (1.8) and (1.3) the theorem can be proved.
ð1Þsþk xs yk 1 ðA þ IÞnþm ½ðA þ IÞsþk s!k!ðn sÞ!ðm kÞ! k¼0
ð1Þsþk xs yk ðA þ IÞnþm ðBÞsþk C1 ðBÞ½ðA þ IÞsþk 1 s!k!ðn sÞ!ðm kÞ! k¼0
which according to the definition of Appell’s series in two variables F1 , yields the required result (3.1). h A special case of the transformation formula (3.1) is worthy of note. Indeed, upon letting B ¼ A þ I, (3.1) readily yields
274 Z
M.G. Bin-Saad
1
ek k A LðA;kÞ n;m ðx; yÞdk ¼
0
C1 ðA þ IÞ ð1 xÞn ð1 yÞm : n!m!
ð3:2Þ
Theorem 3.2. Let A be matrix in CNN satisfying spectral condition ðaÞ is not an eigenvalue of A for every integer a > 0 and RðkÞ > 0. Then Z 1Z 1 exy xBI yCI LðA;kÞ n;m ðx; yÞdxdy 0
0
ðA þ IÞnþm C1 ðBÞC1 ðCÞ F3 ½B; nI;C; mI; B;k; k; ¼ n!m!
Proof. Denote, for convenience, the left-hand side of Eq. (3.1) by I,then ð1Þsþk ksþk ðA þ IÞnþm ½ðA þ IÞsþk 1 I¼ s!k!ðn sÞ!ðm kÞ! s¼0 k¼0 Z 1 Z 1 ex xBþðs1ÞI dx ey yCþðk1ÞI dy ¼
where aðs; kÞ – 0; l; m; d; r 2 C; z ¼ z1 ; . . . ; zr ; w ¼ w1 ; . . . ; wi ; and ½np X ½mq X ðA;kÞ Hl;m;d;r ðx; y; z; w; f; nÞ ¼ aðs; kÞLnps;mqk ðx; yÞXlþms;dþrk n;m;p;q s¼o k¼o
ð4:2Þ NN
whose eigenvalues where n; m; p; q 2 N; A is a matrix in C c; ðcÞ is not eigenvalue of A for every integer c > 0. Then 1 X a b l;m;d;r Hn;m;p;q x; y;z;w; p q un vm u v n;m¼0 kðxu þ yvÞ Kl;m;d;r ðz;w;a; bÞ: ¼ ð1 u vÞðAþIÞ exp ð4:3Þ 1uv
0
s¼0 k¼0
s;k¼0
ðz; wÞfs nk ;
n X m X
0
Theorem 4.1. Corresponding to an identically non-vanishing function Xl;d ðz; wÞ of r complex variables z1 ; . . . ; zr , of complex order l and i complex variables w1 ; . . . ; wi of complex order d; ðr; i 2 NÞ, let 1 X Kl;m;d;r ðz; w; t; lÞ ¼ aðs; kÞXlþms;dþrk ðz; wÞts lk ; ð4:1Þ
ð3:3Þ
where F3 is Applls´ function (see Srivastava and Karlsson (1985)).
n X m X
essentially arbitrary coefficients for the two variable Laguerre matrix polynomials which are generated by (1.3) and given explicitly by (1.2). We begin by stating the following theorem.
ð1Þsþk ksþk ðA þ IÞnþm ½ðA þ IÞsþk 1 s!k!ðn sÞ!ðm kÞ!
CðB þ sIÞCðC þ kIÞ which according to the definition of Appell’s series in two variables F3 , yields (3.2). h
provided that each member of (4.3) exists for ju þ vj < 1 and RðkÞ > 0.
Theorem 3.3. Let A be matrix in CNN satisfying spectral condition ðaÞ is not an eigenvalue of A for every integer a > 0 and RðkÞ > 0 . Then
Proof. Denote, for convenience, the right-hand side of Eq. (4.3) by I. Then from (4.2), we get
Z
I¼ t
1 ðt kÞBþI k A LðA;kÞ n;m ðx; yÞdk ¼ CðBÞCðA þ ðn þ mÞIÞC
ð3:4Þ
Proof. Denote, for convenience, the left-hand side of Eq. (3.1) by I,then I¼
n X m X s¼0 k¼0 Z t
ð1Þsþk xs yk ðA þ IÞnþm ½ðA þ IÞsþk 1 s!k!ðn sÞ!ðm kÞ!
ðt kÞBI kAþðsþkÞI dk:
0
putting t z ¼ tð1 pÞ, we get I¼
n X m X s¼0 k¼0
Z
sþk s k
ð1Þ x y ðA þ IÞnþm ½ðA þ IÞsþk 1 tAþBþðsþkÞI s!k!ðn sÞ!ðm kÞ!
1
ð1 pÞBI pAþðsþkÞI dp;
ðA;kÞ
aðs;kÞLnps;mqk ðx;yÞXlþms;dþrk ðz;wÞas bk unps vmqk ;
n;m¼0 s¼o k¼o
0
ðA þ B þ ðn þ m þ 1ÞIÞtAþB LðAþB;tÞ ðx; yÞ: n;m
½np X ½mq 1 X X
0
which on applying the definition of Beta matrix function (1.6) and considering (1.2), we get (3.4). 4. Bilinear and bilateral generating matrix functions We aim here at presenting a family of bilinear and bilateral generating matrix functions involving multiple series with
which on letting n # n þ ps; m # m þ qk, we can write I¼
1 X
s k n m aðs; kÞLðA;kÞ n;m ðx; yÞXlþms;dþrk ðz; wÞa b u v
n;m;s;k¼0
¼
1 X
n m LðA;kÞ n;m ðx; yÞu v
n;m¼0
!
1 X
! s k
aðs; kÞa b
s;k¼0
kðxu þ yvÞ Kl;m;d;r ðz; w; a; bÞ; ¼ ð1 u vÞðAþIÞ exp 1uv which completes the proof of Theorem 4.1.
h
By appropriate choices of the generating functions in (4.1), we can derive a number of generating functions involving the products of such polynomials as the familiar Laguerre, Gegenbauer, Jacobi and Hermite matrix polynomials. First of all, in its special case when r ¼ i ¼ 1; l ¼ d ¼ 0; m ¼ r ¼ 1; aðs; kÞ ¼ 1and using the generating relation (1.3), the assertion (4.3) would obviously yield the following bilinear generating function: ½np X ½mq 1 X X ðA;kÞ ðB;gÞ Lnps;mqk ðx; yÞ Ls;k ðz; wÞanps bmqk un vm n;m¼0 s¼o k¼o
¼ ð1 u vÞðAþIÞ ð1 a bÞðBþIÞ kðxu þ yvÞ gðza þ wbÞ þ : exp 1uv 1ab
ð4:4Þ
A note on two variable Laguerre matrix polynomials
275
Secondly, consider the generating function (Ghazi, 2008):
1 2ux 2vy þ u2 þ v2
A
¼
1 X
A Cn;m ðx; yÞun vm :
ð4:5Þ
n;m¼0 A ðx; yÞ is Gegenbauer matrix polynomials of two where Cn;m variables, upon setting r ¼ i ¼ 1; l ¼ d ¼ 0; m ¼ r ¼ 1; aðs; kÞ ¼ 1 and A ðz; wÞ; Xs;k ðz; wÞ ¼ Cs;k
in (4.3), we shall obtain a bilateral generating function in the following form: ½np X ½mq 1 X X ðA;kÞ Lnps;mqk ðx; yÞ CBs;k ðz; wÞanps bmqk un vm B
ð4:6Þ
1 X ða þ 1Þnþm xn ðxÞm a;bn a;bm P ðzÞ Pm ðyÞ ða þ 1Þn ða þ 1Þm n n;m¼0
n
x xy x xzoðaþbþ1Þ 1þ 1 þ 2 2 2 2 2 2 3 a þ b þ 1; a þ b þ 1; 6 x2 ð1 zÞð1 yÞ 7 7: ð4:7Þ F1 6 4 ð2 þ x xzÞð2 x þ xyÞ5 aþ1
On ðaþ1Þnþm ðaþ1Þn ðaþ1Þm
letting , taking
r ¼ i ¼ 1; l ¼ d ¼ 0; m ¼ r ¼ 1; aðs; kÞ ¼
ðzÞ Pka;bk ðwÞ; Xs;k ðz; wÞ ¼ Pa;bs s and combing (4.7) with (4.1), we get ½np X ½mq 1 X X n;m¼0 s¼o
1 X ða þ 1Þnþm xn ðxÞm ðrþ1Þ FD ½a; b1 ; . . . ; br ; n; c; z1 ; . . . ; zr ; z n!m! n;m¼0 ðiþ1Þ
Next, consider the double generating function of single polynomials (Pathan and Bin-Saad, 1999):
¼
Finally, we recall here a double generating relation for hypergeometric functions of several variables (see Pathan and Bin-Saad (1999))
FD ½d; e1 ; . . . ; er ; m; f; y1 ; . . . ; yi ; y l 1 X ðaÞl ðdÞl ða þ 1Þl ðx2 yzÞ ðrþ1Þ ¼ FD l!ðcÞl ðfÞl l¼0
n;m¼0 s¼o k¼o
¼ ð1 u vÞðAþIÞ ð1 2za 2wb þ a2 þ b2 Þ kðxu þ yvÞ exp : 1uv
½np X ½mq 1 X X anps bmqk un vm ðA;kÞ ðB;CÞ Lnps;mqk ðx; yÞ Hs;k ðz; wÞ s!k! n;m¼0 s¼o k¼o pffiffiffiffiffiffi ¼ ð1 u vÞðAþIÞ exp 2Bza a2 I pffiffiffiffiffiffi kðxu þ yvÞ 2 : ð4:10Þ exp 2Cwb b I exp 1uv
ða þ 1Þsþk ðA;kÞ Lnps;mqk ðx; yÞ Pa;bs ðzÞ s ða þ 1Þ ða þ 1Þ s k k¼o
ðwÞanps ðaÞmqk un vm Pa;bk k n
a za a waoðaþbþ1Þ 1 þ ¼ 1þ ð1 u vÞðAþIÞ 2 2 2 2 kðxu þ yvÞ exp 1uv 2 3 2 a þ b þ 1; a þ b þ 1; 2 6 a ð1 zÞð1 wÞ 7 7 ð4:8Þ F1 6 4 ð2 þ a zaÞð2 a þ wa5 a þ 1;
½a þ l; b1 ; . . . ; br ; 1 þ a þ l; c þ l; z1 ; . . . ; zr ; xz ðiþ1Þ
FD
½d þ l; e1 ; . . . ; er ; 1 þ a þ l; f þ l; y1 ; . . . ; yi ; xy: ð4:11Þ
ðnÞ
where FD is Lauricella’s function of n-variables (Srivastava and Karlsson, 1985). ðaþ1Þ Upon setting l ¼ d ¼ 0; m ¼ r ¼ 1; aðs; kÞ ¼ s!k!sþk , letting dimensionðrÞ # dimensionðr þ 1Þ and dimensionðiÞ # dimensionði þ 1Þ;
taking Xs;k ðz1 ; . . . ; zr ; z; w1 ; . . . ; wi ; wÞ ðrþ1Þ
¼ FD
½a; b1 ; . . . ; br ; n; c; z1 ; . . . ; zr ; z
ðiþ1Þ FD ½d; e1 ; . . . ; er ; m; f; y1 ; . . . ; yi ; y;
in (4.1) and proceeding in the manner described above it is not difficult to obtain the following multi-variable bilateral generating relation: ½np X ½mq 1 X X ða þ 1Þ
sþk
n;m¼0 s¼o k¼o
s!k!
ðA;kÞ
ðrþ1Þ
Lnps;mqk ðx; yÞ FD ðiþ1Þ
½a; b1 ; . . . ; br ; s; c; z1 ; . . . ; zr ; z FD
½d; e1 ; . . . ; er ; k; f; w1 ; . . . ; wi ; wanps ðaÞmqk un vm kðxu þ yvÞ ¼ ð1 u vÞðAþIÞ exp 1uv l 1 2 X ðaÞl ðdÞl ða þ 1Þl ða wzÞ ðrþ1Þ FD l!ðcÞ ðfÞ l l l¼0 ½a þ l; b1 ; . . . ; br ; 1 þ a þ l; c þ l; z1 ; . . . ; zr ; az ðiþ1Þ
FD
½d þ l; e1 ; . . . ; er ; 1 þ a þ l; f þ l; w1 ; . . . ; wi ; aw: ð4:12Þ
5. Conclusion Similarly, on considering the generating function [1]. 1 pffiffiffiffiffiffi X un vm ¼ exp 2Axu u2 I Hn;m ðx; y; A; BÞ n! m! n;m¼0 pffiffiffiffiffiffi ð4:9Þ exp 2Byv v2 I ; we can show that
In this paper, integral operators and series rearrangement technique has been applied to obtain finite and infinite sums, integral transforms and bilinear and bilateral generating matrix functions for Laguerre matrix polynomials of two variables. Also, some interested particular cases and consequences of our results have been discussed.
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