On mixed trilateral generating relation of generalized Bessel polynomials-II

              © Del documento, de los autores. Digitalización realizada por ULPGC. Biblioteca Universitaria, 2017
Rev. Acad. Canar. Cierre., XII (Núms. 1-2), 19-22 (2000)
ON MIXED TRILATERAL GENERATING
RELA TION OF GENERALIZED BESSEL
POL YNOMIALS - 11
ASHUTOSE SAHA
1. Introduction: In this note we shall adopt group theoretic method
to obtain mixed trilaleral generating relations involving generalized
Bessel Polynomials from the existence of a class of bilinear generating
relations. The main result of this note is stated in the form of the
following theorem.
2. Theorem : If there exists a bilinear generating relation of the form
~
G (x, ;, w) = !: an Y n (a) (x) Y., (a) (;) w"
n~J
tlwn
Pw Pw a 2 a 2 ex p (---- + ---) {1 + wx} - {l + w;} - x
1 + wx 1 + wl; ·
G (x(l + wx), 1; (1 + wl;), vw)
"' "'
= !: !: (wp)P fn,p (x,1;,v,w),
1141 p ~ o
where
min (n.p) a
f ( ~ ) ~ ____ n.:_q_ _ (vw)n-q Y(na+-p2p-2+q2q) (x) yn<a-2q) ( ~) .
n,p x,~,v,w = q"=°'o (p-q) ! q ! ~
Proof : For the generalized Bessel polynomials Y n (al(x), we consider
the following linear partial differential operator [1]
R 2 -2 a -1 a A -2 2 -2
1 = x y z - -- + xy z --- + ,...y z- xy z
ax ay
The work has been done when the author was a research fellow in the
Department of Mathematics, University of Kalyani, India.
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such that
and
(2.2) ewR1 u (x;y,z) = exp [¡3wyº2z/{1 + wxyº2z}] (1 + wxyº2z)"2 x
u (x (1 + wxyº2z), y (1 + wxyº2z), z).
A.lso for the Polynomials Y n (a) (l;) we consider
such that
and
(2.4) ewR2 u (l;,11,I;) = exp [ ¡3w1f2 ~/{1 + wl;11"21;}] {1 + wl;11"21;}"2x
u (l; (1 + wl;11"21;), 11 (1 + wl;11.21;), ¡;).
Nuw consider the generating relati9n
Replacing w by wzl;v and then multiplying both sides of (2.5) by yª11ª,
Wl' gct
(2.h) y"T\u G (x,l;,wzl;v) = yª11ª "Í":'° a n Y n( a) (x) Y n( a) (l;) (wzl;v)n
= Í: ªn (wv)n (Y n (a) (x) yª zn) (Y n (a) (l;) 11ª l;n).
n:{)
Now operating both members of (2.6) by (expwR1) (expwR2), we get
(2.7) (vxpwR1) (expwR2) [yª '1ª G (x, ¿;, wzl;v)] =
(expwR1) (expwR2)[Í:an (wv)n (Yn(a) (x) yª zn) (Y (a)(¿;) '1ª/;n)] .
n:{) n
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Left hand side of (2.7) with the help of (2.2) and (2.4) becomes
(2.8) (expwR1) (expwR2) [yª 11ª G (x, ~' wz~v)]
= (expwR1) exp [¡3w11·2 ~ / {1 + w~11-2 ~)] (1 + w~11-2 ~)"2 x
/XT\IX (1 + w;r¡-2 s)ª G (x,; (1 + w;11·2 s), wzsv)
= cxp [~wy-2z/{1 + wxy-2z}](l + wxy-2zr2x
exp [~wr¡·2 ~ / {1 + w;r¡-2 ~}] (1 + w;11·2 ~)"2
ycx {1 + wxy"2 z)ª11ª (1 + w;11·2 ~)ª
G (X (1 + wxy"2 z),; (1 + w;1f2 s), wz~v).
Also the right hand side of (2.7) with the help of (2.1) and (2.3) becomes
(2.9)
TherdorL· equatíng (2.8) and (2.9), wc gct
(2.10) exp [¡3wy"2z / {1 + wxy"2z}] (1 + wxy-2z)"2x
CXp [~Wlf2 ~ j {1 + w;1f2s}] (1 + W;l'\-2s)"2 yª (1 + WXy-2z}IX X
¡;ir_ (,)le °'" wP•q = L L L [a 0 (wv)º ---- (~P Y n+ (cx -ip) (x) yª"2P zn+p) x
11-"ll p-'4) q~ p! q! p
Now putting y=z= 11=~=l in (2.10) we get
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exp [pw/(l + WX)) (1 + wxr2 exp (~w/(l + W~)) (1 + w~r2 X
(1 + wx)ª (1 + w~)ª G (x (1 + wx), ~ (1 + w~), vw)
~ ~ ~ [· ( )n -~~~- rtr+q y (n· 2p) (x) y (1t·2q) (~)J.
- ¿,... ¿,... ¿,... ,111 WV 1 1 1-' n+p n+<¡ ~
11=1! p:O •1~0 p. q.
Simplifying we obtain
exp (--P-w- -- + --P-w- --) (1 + wx) a-2 ( 1 + w~~) a-2
1 + wx 1 + w;
G (x (1 + wx), ~ (1 + w~), vw)
= .L. L~ minL (n ,p) a (Aw)P (a 2 +2 ) (vw)n-q _ _y_ ___ Y - P q (x) Y <u·2q) (~).
n=ü p=O q=O n-q (p-q)! q ! n+p-2q n
This completes the proof of the theorcm.
Acknowledgement: The author is gratcful to Dr. A. K. Chongdar,
Depa rtmcnt of mathematics, Bangbasi Evening College, for his kind
help in the preparation of this paper.
Reference:
A. K. CHONGDAR: On G~nerating Functions of Bessel Polynomials
by Group-Theoretic Method, Ballettino U.M.I. (7) 5-A (1991), 163-170.
Department of Mathematics
Bangladesh Institute of Technology
Chittagong, Bangladesh.
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