Journal of Pure Mathematics, Vol-26

2009·6.2 MB·

Volume 26, 2009
* v. S
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JOURNAL OF PURE MATHEMATICS
UNIVERSITY OF CALCUTTA
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\\ t-V
Jour. Pure Math., Vol. 26, 2009, pp. 1-13
ON SOME MULTIVARIABLE GENERALIZED
TRUESDELL POLYNOMIALS
R. C. Singh Chandel & S. S. Chauhan
ABSTRACT : Motivated by the work of Chandel and Tiwari [1] and Singh [2], here in the present
paper we introduce and study multivariable generalized Truesdell polynomials defined by (1.3). Its further
generalization will also be discussed.
Key words : Generalized Truesdell polynomials, multivariable analogues, Generating relations, Rodrigues
formula. Pure Recurrence relations, Differential Recurrence Relations.
2000 Mathematics Subject Classification. 33C65, Secondary 33C70.
1. INTRODUCTION
Chandel and Tiwari [1] introduced multivariable analogue of Hermite polynomials defined by
Rodrigues’ formula :
where nv.....,nm are positive integers, while hx,...,hm, are any numbers real or
complex independent of Xj,....,xm.
Singh [2] introduced generalized Truesdell polynomials by the Rodrigues’ formula :
(1.2)
where n is positive integer and a, p, r are arbitrary numbers real or complex independent
of x.
Motivated by the above work [1,2], here in the present paper, we introduce and study
2 R. C. SINGH CHANDEL & S. S. CHAUHAN
multivariable generalized Truesdell polynomials defined by
Rodrigues’ formula:
= l-(ablogjc, ,..-(aOT logxm -pmxr°')
5? • • • 5IT [l - (a, log x, - pxx?(a* log xm - pmxr”)
where nt are positive integers, b, ae rf pi are arbitrary numbers real or complex independent
of x and S, a x, ;i = 1,....m.
It is clear that
fi-Ei Em-n r -EL
J b’-’ b b'’ b ), s
(1-4) ...'(*.............................................*4
7^lixl>ruPl)--7Zm{Vm^Pm)
=
where I^(x, r, p) are generalized Truesdell polynomials of Singh [2] defined by (1.2).
2. SOME FAMILIAR PROPORTIES OF 5 = x-£~.
ox
Here we write some familiar properties of 5, which will be useful in our investigations;
(2.1) §"{*«} = anx2,
(22) 5”{w.v}= Z (|)5',-i{w}8*{v},
k=0 v '
(2.3) F(5){e*W/(x)} = ^F(d + xg'){/(x)}
and
(2.4) e*m} = fee*).
ON SOME MULTIVARIABLE GENERALIZED TRUESDELL POLYNOMIALS 3
3. GENERATING RELATION.
Starting from defination (1.3), we have
rJb.ai. ,r„,.ph..pm}f \ fm"
, "1. .nm \x\> — ’xm) y—n j
1 - (a, log xi - pxxP )-•••- (aw logxm - pmx%)
e+(/,81+ +'«8«)i|1 _ Jttl logjfj - )-•••- (a,„ logxm - pmx%)
Now making an appeal to (2.4), we derive the generating relation:
nl ,nn
(3.0 ? .......
nx\'"nm\
”1. ••nm=°
= [l - («! log x{ - pxx?(aw log xm - pmxr™)
-b
1 -{ai(logXj + tx)~ pxx^e^- ...-[am{\ogxm+tm)~ p^’e^]
4. APPLICATIONS OF GENERATING FUNCTION.
Through generating function (3.1), we derive
Jb+b'-,ax,...,am-,rx,..sm-,px,.„,pm)( \
(4.1)
— Y X5 / rtt \ (nm\ji^ai....am:ri....rm'P\?-'Pm){ „ \
( m)
l’-’
rAb ,o.x,...,<xm,i\,.~,r'm,px,—,pm)/
\
4 R. C. SINGH CHANDEL & S. S. CHAUHAN
5. PURE RECURRENCE RALATIONS
Differentiating (3.1) partially with respect to /, and equating the coefficients of t"1 both
the sides, we obtain pure recurrence relation:
1 - (a, log*, - p,*?) - ... - (am log*OT - pmx%)
(5.1)
— hrt 7-(6+ha 1am'A,Sm*P\’-'Pm)\
\xh->xm)
-bLpnin_ xi> 1 Ln .... ....V V
r°>'Px W’-’W
which suggests m results in the following unified form :
(5.2) [l “ («i log*, - P,*,n)-■••- (a* log*^ - pmx%)
r(6;a1,...,am;a1,._,am;q,_..rm)/ \
lnu...,n,.x,n,+\,n,^.,nm
= bailf+V,*u'",am'A...rm’Px’-,Pm\xh...,xm)
»***»'7n x
—hn,rxr' Y r^‘ (n‘ ....rm<Pl'-'Pm) f \.
fmxi lr? ...KxU-^mh (i = 1,..., m)
k=0
6. DIFFERENTIAL RECURRENCE RELATIONS.
Differentiating (3.1) partially with respect to x{ and equating the coefficients of r"1 ...r”” , we
establish
.1) £l - (a, log*, - p,*?) - ... - (am log*m - Ptnxm )
(6
ON SOME MULTIVARIABLE GENERALIZED TRUESDELL POLYNOMIALS 5
d rAb,0
\,—\
dxx
^ I’‘"’
= .......
jCi
■dwf15 ..'‘■W-*-)
*=o
which can be furtht written as
(6.2) 1 - (a i .og xi - pxx?) - ... - (am log xm - pmx%)
2 jib’ai’r-’am’rl’-’rm’Pl-.’Pm) (v v \
.....rz,, \xh-"’xm) ■
= -h(a, - ......
K~~ U
This result further suggests m differential recurrence relations in the following unified form:
(6.3) [l - («i log*i - P\x^] - ... - (aOT log*m - pmx%)
s >rib',a.i,...,am',r\,...,rm',pi,...,pm)/ \
0/-L r
m
= -A(ar-^)^;“'-a“:n....w"-p>n\xh...,xm)
+bax4t+!Z'' , ,r",;Pl....’Pm)(*l > • • ■ * *«)
-hnrr* ? (ni]rk'Ab+l;ai....a*':r'....r":p1....p“)fx, X )
-bptnxt i u yt ^_M+l...% (X],...,Amj z = l,...,/77.
fc=0
6 R. C. SINGH CHANDEL & S. S. CHAUHAN
An appeal to (5.2) and (6.3) further shows that
1 - (aj log*! - pxxr()-...- (ct/n logxm-pmx%)
(6.4)
1n[t„,nl_un,+[,nt+h„,nm \ m)
- 1 - (a, log*, - Plx? )-•••- (aw logxm - pmx%)
)iiTX'a’"') vn
pivi
- -*(“-■ - .... ......................*»);' -1
which can be further written as
b(pinxi‘ - «/) + 1 - (“1 Iog*i - P\X^ )-•••- («m logX„
(6.5) Pmxm
/7j ' fft /
1 - (aj log*! - prf(am logxnl - Pmxfa)
rAb,CL\,—,a.m,i\,—,fmtP\’—>Pm)(T \ ■ _ i m.
inu...,n,_l,n,+\,n,+u...tnm \x\»-*xm) 1 W-*
that is
b{pinxi‘ -<*,)
■ + 5,-
(6.6)
I - (a,- log*! - Pix[‘) - ... - (a„ logxm - pmx%)
rrib,0.l,~.y0.myt^i--y \
nh.-.nm \xh-’xm)
_ Jb;a,._..a„;/i rm>Pl Pm)(Y v \ ; _ 1
ON SOME MULTIVARIABLE GENERALIZED TRUESDELL POLYNOMIALS 7
Now for brevity, we consider the operator
b[pirixi' - a,)
(6.7) © = + 8,- (/ l,...,m).
- (ai log*i ~ P\x^(aOT log*m - Pmxr«)
Therefore, (6.6) reduces to
(6.8)
"lA+hn,+i,...nnl Kxl’->xmh (z
which further shows that
(6.9) ©,* 1*-•«») (jf,,..., X)n)
~ Jn1,..^l,rh+k,nl+l^nm (xl>-’xm) (*=1 .-.»»)•
Specially for k=nl(j=\,...,m), (6.9) gives
(6.io) ...“..."■W-.*)}
_ ia«|illv"4lP||- ’Pm) ( v y. \ /. , . , V
Also
(6.11)
/=!
Also from (6.9), we derive
J<D, \ T{b;al,.~,am;ri,...,rn-,pl,...,pm)/ \
(6.12) e |J«|.....nm
V x
*! ...."m
R. C. SINGH CHANDEL & S. S. CHAUHAN
7. FURTHER GENERALIZATION.
(7.0 ...............
(g[(“i lr|g - m') + ■ + («», logxm - pi4”))
G CCl logx, - PiX + ... + (am !ogxm-Plx%')}
where
(7.2) G(z) = Hlnzn, To ^ 0
n=0
(£)
which for yn = reduces to (1.3).
white for yn =-i—i— (7.1) defines the multivariable polynomials as
rn on r'(al'—’Clnr,I>—>7»>Pl-—jPra)/'v v \
= exp(pi^ri+-+^m K’-”8S'm
m
It is quite clear that
(7.4) ...P.\Xh...,Xm)
^{xUn,P\)-Tt”’{Vm,Pm\
=
where T^(x,r,p) are eneralized truesdell polynomial of Singh [2]