IS S N 1 3 0 3 –5 9 9 1
NEW GENERATING FUNCTIONS FOR THE KONHAUSER MATRIX POLYNOMIALS
ESRA ERKU¸S - DUMAN AND BAYRAM ÇEK·IM
Abstract. Varma et. al. [Ars Combin. 100 (2011) 193-204] introduced the concept of the Konhauser matrix polynomials. In this paper, we obtain some generating functions for these matrix polynomials. Finally, we focus on some special cases.
1. Introduction
In 2011, Varma et. al. de…ned the pair of the Konhauser matrix polynomials as follows: Zn(A; )(x; k) = (A+(kn+1)I)n! n X r=0 ( 1)r n r 1(A + (kr + 1)I)( x)kr; (1.1) Yn(A; )(x; k) = 1 n! n X r=0 ( x)r r! r X s=0 ( 1)s r s 1 k((s + 1)I + A) n ; (1.2) where A is a matrix in CN N satisfying the condition
Re( ) > 1 for every 2 (A); (1.3)
is a complex number with Re( ) > 0 and k 2 Z+ (see [5]): Here, for any matrix
A in CN N; Pochhammer symbol is de…ned by
(A)n= A(A + I):::(A + (n 1)I); n 1 ; (A)0= I:
They show that the Konhauser matrix polynomials Zn(A; )(x; k) and Yn(A; )(x; k)
are biorthogonal with respect to the weight matrix function xAe x: Furthermore
Received by the editors Nov. 18, 2013, Accepted: May 15, 2014.
2000 Mathematics Subject Classi…cation. Primary 33C45 ; Secondary 15A60.
Key words and phrases. Konhauser matrix polynomials, multilinear generating matrix func-tion, multilateral generating matrix funcfunc-tion, jacobi matrix polynomials.
c 2 0 1 4 A n ka ra U n ive rsity
they derive the following generating matrix functions: (1 w) 1k(A+I)exp h xn(1 w) 1k 1 oi = 1 X n=0 Yn(A; )(x; k)wn ; jwj < 1 (1.4)
for the polynomials Yn(A; )(x; k) and
(1 t) B 1Fk 2 6 6 4 B ; t( x) k (1 t)kk 1 k(A + I) ; :::; 1 k(A + kI) 3 7 7 5 = 1 X n=0 (B)nZn(A; )(x; k) [(A + I)kn] 1tn ; jtj < 1; t( x)k (1 t)kk < 1 (1.5)
for the polynomials Zn(A; )(x; k); where A and B are matrices in CN N satisfying
the conditions
Re( ) > 1 for every 2 (A)
AB = BA: (1.6) and1Fk is de…ned as 1Fk 2 4 B ; x A1 ; :::; Ak 3 5 =X1 n=0 (B)n n! [(Ak)n] 1 ::: [(A1)n] 1xn
for A1; :::; Ak and B are matrices in CN N satisfying condition Ai+ sI is invertible
for s 2 N, i = 1; :::; k;see [5]. In the present paper, we obtain multilinear and multilateral generating functions for the pair of the Konhauser matrix polynomials. Some special cases are also given.
2. multilinear and multilateral generating matrix functions In this section, we give theorems which derive several substantially more general families of bilinear, bilateral generating functions for the Konhauser matrix polyno-mials de…ned by (1.1) and (1.2). Using the similar method considered in [1, 2, 3, 4], we obtain the main theorems.
Theorem 2.1. Corresponding to a non-vanishing function (y1; :::; ys) of
com-plex variables y1; :::; ys(s 2 N) and of complex order , let ; (y1; :::; ys; z) :=
1
X
k=0
and
n;p; ; (x; y1; :::; ys; ) := [n=p]X
l=0
al Yn pl(A; )(x; k) + l(y1; :::; ys) l ; (2.2)
where A is a matrix in CN N satisfying the conditions in (1.6); n; p 2 N and (as
usual) [ ] represents the greatest integer in 2 R: Then, for jtj < 1 and Re( ) > 0; we have 1 X n=0 n;p; ; x; y1; :::; ys; tp t n (2.3) = (1 t) 1k(A+I)exp h x n (1 t) 1k 1 oi ; (y1; :::; ys; ):
Proof. For convenience, let S denote the …rst member of the assertion (2.3) of Theorem 2.1. Plugging the polynomials
n;p; ; x; y1; :::; ys;
tp ;
which comes from (2.2) into the left-hand side of (2.3), we obtain
S = 1 X n=0 [n=p]X l=0 al Yn pl(A; )(x; k) + l(y1; :::; ys) ltn pl: (2.4)
Upon changing the order of summation in (2.4), if we replace n by n + pl; we can write S = 1 X n=0 1 X l=0 alYn(A; )(x; k) + l(y1; :::; ys) ltn = 1 X n=0 Yn(A; )(x; k) tn ! 1 X l=0 al + l(y1; :::; ys) l ! = (1 t) 1k(A+I)exp h xn(1 t) 1k 1 oi ; (y1; :::; ys; );
which completes the proof of Theorem 2.1.
Theorem 2.2. Corresponding to a non-vanishing function (y1; :::; ys) of
com-plex variables y1; :::; ys(s 2 N) and of complex order , let
; (y1; :::; ys; z) := 1
X
k=0
and n;p; ; (x; y1; :::; ys; ) (2.6) = [n=p]X l=0 al + l(y1; :::; ys)(B)n plZn pl(A; )(x; k) (A + I)k(n pl) 1 l ; where A and B are matrices in CN N such that A satis…es condition in (1.6); n; p 2
N; and (as usual) [ ] represents the greatest integer in 2 R: Then, for jtj < 1; t( x)k
(1 t)kk < 1 and Re( ) > 0, we have 1 X n=0 n;p; ; x; y1; :::; ys; tp t n (2.7) = ; (y1; :::; ys; )(1 t) B 1Fk 2 6 6 4 B ; t( x) k (1 t)kk 1 k(A + I) ; :::; 1 k(A + kI) 3 7 7 5 : Proof. The proof is similar to Theorem 2.1.
Now, we obtain some special cases for generating functions.
Firstly, if we set + k(y ) = P(C;D)+ k (y) ( ; 2 N0) for s = 1 in Theorem 2.1,
where the Jacobi matrix polynomials Pn(C;D)(y) are de…ned by means of the
gener-ating function in [7]: 1 X n=0 Pn(C;D)(x)tn = F4 I + D; I + C; I + C; I + D; (x 1) t 2 ; (x + 1) t 2 (2.8) r (x 1) t 2 + r (x + 1) t 2 < 1 ! ;
where C and D are matrices in CN N satisfying the spectral conditions Re(z) > 1
for each eigenvalue z 2 (C); and Re( ) > 1 for each eigenvalue 2 (D), CD = DC and F4(A; B; C; D; x; y) is de…ned by
F4(A; B; C; D; x; y) = 1 X n;k=0 (A)n+k(B)n+k(D)n1(C)k1x kyn k!n!;
where C + nI and D + nI are invertible for every integer n 0 in px + py < 1: Then we obtain the following example which provides a class of bilateral generating functions for the Jacobi matrix polynomials and the Konhauser matrix polynomials Yn(A; )(x; k).
Example 2.3. Taking al= 1; = 0; = 1 and jtj < 1; we have 1 X n=0 [n=p]X l=0 Yn pl(A; )(x; k) Pl(C;D)(y) ltn pl = (1 t) 1k(A+I)exp h xn(1 t) 1k 1 oi F4 I + D; I + C; I + C; I + D; (y 1) 2 ; (y + 1) 2 ; where q (y 1) 2 + q (y+1) 2 < 1:
If we take + k(y ) = CD+ k(y) ( ; 2 N0) for s = 1 in Theorem 2.2, where
the Gegenbauer matrix polynomials CD
n(y) are de…ned by means of the generating
function in [6]: 1 X n=0 [(2D)n] 1CnD(x)tn= exp (xt) 0F1 ; D + I 2; 1 4t 2 x2 1 ;
where D is a matrix in CN N satisfying the spectral condition z
2 2 (D) for=
each eigenvalue z 2 Z+[ f0g ; then we obtain the following example which provides
a class of bilateral generating functions for the Gegenbauer matrix polynomials CD
n(x) and the Konhauser matrix polynomials Z (A; ) n (x; k). Example 2.4. Taking al= [(2D)l] 1 ; = 0; = 1; we have 1 X n=0 [n=p]X l=0 [(2D)l] 1 ClD(y)(B)n plZn pl(A; )(x; k) (A + I)k(n pl) 1 l tn pl = exp (y ) 0F1 ; D + I 2; 1 4 2 y2 1 (1 t) B 1Fk 2 6 6 4 B ; t( x) k (1 t)kk 1 k(A + I) ; :::; 1 k(A + kI) 3 7 7 5 ; where A and B are matrices in CN N such that A satis…es conditions in (1:6):
Substituting + k(y ) = Y(C;+ k2)(y; k2) ( ; 2 N0) for s = 1 in Theorem 2.2,
then we can give the following example which provides a class of bilateral generating functions for the pair of the Konhauser matrix polynomials.
Example 2.5. Taking al= 1; = 0; = 1 and jtj < 1; we have 1 X n=0 [n=p]X l=0 Y(C; 2) l (y; k2)(B)n plZ (A; ) n pl(x; k) (A + I)k(n pl) 1 l tn pl = (1 ) k21(C+I)exp h 2y n (1 ) k21 1 oi (1 t) B 1Fk 2 6 6 4 B ; t( x) k (1 t)kk 1 k(A + I) ; :::; 1 k(A + kI) 3 7 7 5 ; where C is a matrix in CN N satisfying the condition
Re( ) > 1 for every 2 (C);
2is a complex number with Re( 2) > 0, k22 Z+ and j j < 1:
Also setting + l(y ) = Y(C;+ l2)(y; k2) ( ; 2 N0) for s = 1 in Theorem 2.1, we
have the following example which provides a class of bilinear generating functions for the Konhauser matrix polynomials Yn(A; )(x; k).
Example 2.6. Taking al= 1; = 0; = 1 and jtj < 1; we have 1 X n=0 [n=p]X l=0 Yn pl(A; )(x; k) Y(C; 2) l (y; k2) ltn pl = (1 t) k1(A+I)exp h x n (1 t) k1 1 oi (1 ) k21(C+I)exp h 2y n (1 ) k21 1 oi where C is a matrix in CN N satisfying the condition
Re( ) > 1 for every 2 (C);
2is a complex number with Re( 2) > 0, k22 Z+ and j j < 1:
Similarly, in Theorem 2.2, if we take + l(y ) = Z(C;+ l2)(y; k2) ( ; 2 N0),
where C is a matrix in CN N satisfying the condition
Re( ) > 1 for every 2 (C);
2 is a complex number with Re( 2) > 0 and k2 2 Z+ ; then we obtain bilinear
generating functions for the Konhauser matrix polynomials Zn(A; )(x; k):
Notice that, when the multivariable function + k(y1; :::; ys); (k 2 N0; s 2 N);
is expressed in terms of several simpler functions of one or more variables, then each suitable choice of the coe¢ cients ak (k 2 N0) in Theorems 2.1 and 2.2 can be
shown to yield various classes of multilateral and multilinear generating functions for the Konhauser matrix polynomials de…ned by (1.1) and (1.2).
References
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Current address : Gazi University, Faculty of Science, Department of Mathematics, Teknikokullar TR-06500, Ankara, TURKEY
E-mail address : eduman@gazi.edu.tr, bayramcekim@gazi.edu.tr
