Centro Ricerehe Frascati--C.P. 65-00044 Prascati--Roma, Italia. M. X. HE. Department of Mathematics, Nova Southeastern University. Ft. Lauderdale, FL 33314, ...
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MATHEMATICAL AND
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COMPUTER
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Mathematical and Computer Modelling 42 (2005) 1263-1268
ELSEVIER
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Eigenfunctions of Laguerre-Type Operators and Generalized Evolution P r o b l e m s G. DATTOLI ENEA--Unit5 Tecnico Scientifiea Tecnologie Fisiche Avanzate Centro Ricerehe Frascati--C.P. 65-00044 Prascati--Roma, Italia
M. X. HE Department of Mathematics, Nova Southeastern University Ft. Lauderdale, FL 33314, U.S.A.
P. E. R i c c i Universith. di Roma "La Sapienza", Dipartimento di Matematica P.le A. Moro, 2-00185 Roma, Italia
(Received and accepted danuary 2005) consider eigenfunctions of a class of differential operators generalizing the Laguerre derivative. Applications in the framework of generalized evolution problems are also derived. @ 2005 Elsevier Ltd. All rights reserved. Abstract--We
Keywords--Eigenfunctions, Laguerre derivative, Laguerre-type evolution problems.
1. I N T R O D U C T I O N The Laguerre derivative, denoted in the following by DL, and defined by d
d
DL := D x D = -~xX-~x ,
(1.1)
is a special differential operator appearing quite frequently in mathematical modelling relevant to vibrating phenomena in viscous fluids and even in mechanical problems such as the oscillating chain (see [1, pp. 282-284]). In preceding articles, we have shown the role of the Laguerre derivative in the framework of the so called monomiality principle [2] and its application to the multidimensional Hermite (Hermite-Kamp6 de F6riet or Gould-Hopper polynomials, [3-5]) or Laguerre polynomials [2,6,7]. It is easily seen, by induction, that the Laguerre derivative verifies
( D x D ) ~ = D ~ x ~ D n.
(1.2)
In a preceding article [8], we considered the Laguerre-type exponentials,
es(x) :=
k=o (h!) s + l '
0895-7177/05/$ - see front matter @ 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2005.01.034
(1.3)
Typeset by ~42~4S-TEX
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which are eigenfunctions of the iterated Laguerre-type operators,
D(~-I)C : = D x D z . . . D x D -
-
- D (xD + x2D 2 + . . . + x~-ID~-I)
S(s, 1)D + S(s, 2 ) z D 2 + . . . + S ( s , s ) x ~ - l D ~,
(1.4)
where the S(s, h), (h = 1, 2 , . . . , s) are Stirling numbers of second kind [9]. Equation (1.2) can be easily generalized in the form,
( D z D x . . . DxD) ~ = Dnx'~Dnz ~ . . . D'~xnD n.
(1.5)
In the same article, we have shown applications to the solution of Laguerre-type evolution problems, mentioning also suitable extensions to the case of functions which are essentially the multi-index Bessel functions discussed in [10], defined as follows, oo
Xk
(1.6) k=O
In this article we extend the results of [8] considering the case of generalized forms of the eigenfunctions (1.3), for which the relevant differential operator can be easily constructed. The indices of eigenfunctions (1.6) are suitably changed, in order to be written in a more compact form. In the above referred articles, several results were obtained by using a linear differential isomorphism (and its iterations), denoted by the symbol T := T~, acting onto the space A := Ax of analytic functions of the x variable, by means of the correspondence,
D ~ DL := DxD;
x. ~ D~ 1,
(1.7)
where Dx~(1) := z'~/n!. By using such an isomorphism, several generalization of classical special polynomials and applications were considered (see [11-15]). 2.
EIGENFUNCTIONS LAGUERRE-TYPE
OF GENERALIZED DERIVATIVES
We consider in this section operators of the kind
Dh+lxJD j-h,
h, j C N o = N U { O } ,
j>h.
(2.1)
Note that, as particular cases, when h = 0, we find the operator DxJD j, and when j = h + 1 = ~, the operator DexeD. We prove the following result. THEOREM 2.1. The operator (2.1) admits the eigenfunction, oo
e(h+l'J'J--h)(X) :----E k=o
Xk
(~[)2(~ __ 1)!(~ -- 2)!''" (~ -- h - j -~ 1)[(k + h)[(k + h - 1)[.--(k + I)!"
Note t h a t in the denominator of the right-hand side of the last equation, (k!) 2 is multiplied by a total of (j h - 1) + h = j - 1 factorials. PROOF. It is a straightforward consequence of equation, Dh+ixJDJ-hx
k =
k2(k -
l)(k - 2)... (k + h - j + l)(k + h)(k + h - i)... (k + 1)x k-l. (2.2)
Eigenfunctions of Laguerre-Type Operators
1265
REMARK 2.1. In t h e series e x p a n s i o n of t h e eigenfunction of T h e o r e m 2.1, t h e s u m m a t i o n i n d e x satisfies t h e a c t u a l c o n d i t i o n k _> j - h - 1. However, we prefer to s t a r t from k = 0 a s s u m i n g t h e usual c o n v e n t i o n 1/s! = 0, for s < 0. T h e s a m e c o n v e n t i o n will be used in t h e following in o r d e r to avoid explicit expressions for lower b o u n d s of t h e r u n n i n g index. E q u a t i o n (2.2) shows t h a t t h e o p e r a t o r (2.1) is a lowering shift o p e r a t o r for powers x k, in t h e sense t h a t Dh+lxJDJ-hxk
= ck;h,jz k - 1 ,
ck;h,j = const.
(k > j -- h -
1).
(2.3)
C o n s i d e r now t h e p o s i t i v e integers Jl > J 2 " ' " > J~, a n d d e n o t e b y J~l, J ~ 2 , . . . , J ~ a r e a r r a n g e m e n t of t h e s a m e integers a c c o r d i n g to t h e p e r m u t a t i o n , Sl, s2,. • •, s~. THEOREM 2.2.
T h e o p e r a t o r D x j ~ DJ.~ . . . xJ.~,. D j,,. a d m i t s t h e e i g e n f u n c t i o n ,
xk e(1,j81,j~l ..... j,,,,js,.)(2c) : =
k=0 (]c!)r-Fl[fl]r × [ f 2 ] r - 1 × "'" X F r '
where F I , F 2 , . . . , F~ a r e g/yen b y F1 : : (k - 1)!(k - 2 ) ! . . . (k - Jl ÷ 1)!, /;'2 : = (k -- jl)!(]c -- j l -- 1 ) [ . . . (k -- J2 -F l)[,
F~ : = (k - j ~ _ ~ ) ! ( k - j ~ _ , - 1 ) ! . . . (k
j,, + ~)!.
PROOF. T h e result is a consequence of e q u a t i o n , D x J ~ D J ~ . .. x J ~ DJ~,-x k
=k r (k-1)(k-2)...(k-jl+l)
(k-1)(k-2)...(k-j2+l)...
(2.4)
• .. (k - 1)(k - 2 ) - - . (k - j~ + 1)x k-1. In a s i m i l a r way, c o n s i d e r i n g t h e positive integers ~1 -< ~ ' " _< ~ , a n d d e n o t i n g a r e a r r a n g e m e n t of t h e s a m e integers b y fs~, gs2, • • •, gs~, a c c o r d i n g to t h e p e r m u t a t i o n , Sl, s2, • • •, s~, we can prove t h e following result. THEOREM 2.3.
T h e o p e r a t o r D e ~ x e-~, . . . De~,,z e.~ D a d m i t s t h e e i g e n f u n c t i o n , oo
xk
~=0 ( k t ) ~ + ' [ a , F
× [G~F - 1 × . . . × a ~ '
w h e r e G1, G2, . . . , Gr are given b y
G1 : = (k + 1)!(k + 2 ) ! . . . ( k
+ g l - 1)!,
C2 : = (]~ -F el)!(]g -- L01 _L 1 ) [ . . - ( k + e2 - 1)!,
O r : = (]¢ -F er-1)[(]g -F e r _ l Jr- 1 ) ! . - - ( k +g~ - 1)!. T h e results of T h e o r e m s 2.2 a n d 2.3 can be g e n e r a l i z e d as follows.
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THEOREIVl 2.4. D e n o t e b y P l , P2, . . . , P~+ I , a n d ql , q2, . . . , qr, p o s i t i v e i n t e g e r s , s u c h t h a t P l 4. P2 4. • • • 4- P r + l = ql + q2 4- " " " 4- qr + 1. Then, the operator,
(2.5)
D pl x ql D P 2 x q~ .. • D P " x q " D P " + I , admits the eigenfunction, oo
e(p,,,,~,p~,~ .....,~,,~,,p,+~)(x) := ~
k=O
xk
111 x n2 x ... x I]:,.'
w h e r e HI, I I 2 , . . . , IIr are g i v e n b y
1-[, : = k! (k - 1)! (k - 2 ) ! . . .
(k - p,-+t + 1)!,
1-[2 :-- (k -- Pr+t 4- qr)! (k -- P r + l 4- q< -- 1 ) ! . . . ( k - P r + l 4- qr -- Pr 4- 1)!,
n~:=
k-~pj+
q~ ! k - Z p j +
j=2
j=2
q~-I
!...
~-Z;~+
£=1
j=l
q~+l
!.
£=1
REMARK 2.2. Note that the notation we used here is more compact t h a n the one we introduced for the multi-index Bessel functions in equation (1.6). In particular, for the Laguerre-type exponentials, we have the identities, e 2 (3;) ~- e(1,1,1 ) (x), and
e3(:L') ~ 6(1,1,1,1,1 ) (X),
...,
so oil.
REMARK 2.3. Note that, introducing, for shortness the notations, ~ x = V P l x q l D p2xq . . . .
DP~xq~ Dp.,.+,
and 9(x) = e(pl,q,,p2, q...... pr,q,,,p,,+l) (x), the following eigenvalue property is satisfied,
(~¢(ax) = a~(ax).
3. A P P L I C A T I O N S TO G E N E R A L I Z E D LAGUERRE-TYPE EVOLUTION PROBLEMS In this section, we apply the preceding result to the solution of generalized Laguerre-type evolution problems. The possibility of obtaining such applications is based on the following theorem generalizing a known result recalled in [8]. THEOREM 3.1. L e t ~ x b e a d i f f e r e n t i M o p e r a t o r w i t h r e s p e c t t o t h e x v a r i a b l e , a n d d e n o t e b y ~(x) an eigenfunetion oyez,
such that
O~¢(ax)
= a¢(ax),
K
lim x - H e ( x )
-- 1,
(3.1)
x~O+
where H and K d e n o t e p o s i t i v e c o n s t a n t s , t h e n t h e e v o l u t i o n p r o b l e m , ~ x S ( x , t) = ~S(:c,O
K
lim x-HS(x,t)= ~(t),
x~O+
t),
in the half-plane x > O,
(3.2)
Eigenfunctions of Laguerre-Type Operators
1267
where s(t) denotes an analytic function, admits the operational solution,
s(~,t)=¢
(xN° ) s(t).
(3.3)
PROOF. T h e e i g e n f u n c t i o n p r o p e r t y of fJ, implies
(xO)s(t)=
0
(xO)s(t)=~S(x,t),
since O c o m m u t e s w i t h ~ ( x ° ) . F u r t h e r m o r e , as a consequence of h y p o t h e s i s (3.1)2 t h e b o u n d a r y c o n d i t i o n , for x ~ t r i v i a l l y satisfied.
0, is
Accordingly, we can s t a t e t h e following results. THEOREM 3.2.
The evolution problem, D~h+l x j Dzj--d S ( x , t )
DtS(x,t),
in t h e h a l f - p l a n e x > 0,
/r( lilII xh--j+ls(32, t) = $(t), x---*0+
w h e r e / ( : [ ( j - h - 1)]]2(j - h - 2)!(j - h - 3 ) ! . . . 0 ! ( j - 1 ) l ( j - 2 ) ! . . . (j - h)!, and s(t) denotes an anaIytic function, admits the operational solution,
S(x,t)
: e(h+l,j,j_h)(.Tnt)s(t) :
~
xkDkts(t)
k:j--h-1
(]~[)2(k -- l)! " ' ' (~ -~- h -- j -[- 1)[(k -I- h ) ! - - - (/1c -[- 1)!"
M o r e generally, using t h e n o t a t i o n s of T h e o r e m 2.4, we have t h e following. THEOREM 3.3. Let s(t) be an analytic function. Then, the evolution problem
DPlxq~DP~:c"~ ... DP~xq~DP"+~S(x, t) = D t S ( x , t),
K lira ~ - H S ( x , t ) : x----*O+ r+l
in the half-plane x > 0,
~(t),
r
where H := maXh=l,2 ...... + l [ ~ j = h PY -- ~e=h qe -- 1], (we assume that the s u m of q-quantities is omitted when h = r + 1), K : = [H1 × [I2 x - . - × IL]k=H, admits t h e operational solution,
s(x,t) = ~(.1,q,,p~,q2..... p,,q,.pr+~) (xDt) s(t) x k Dkts(t) k=HA-"II1 x H2 x - - - x I L ' where Yi1, I 1 2 , . . . , 1-It a r e given in Theorem 2.4.
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