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Intm. J. Comrytet Marrr., Vol. 78, pp. 303-321 @ 2001 OPA (Ovtrsff PublishereAssociation) N.9.

Reprints available directly ftom the publtuhff Published by lienre uder'

Photocopying pmitted by liense only the Gordon and Bmch Sciene publishG impdrrt,

a member of the Taylor & Fmncis Group.

SOME INTEGRALS OF THE PRODUCTS

OF LAGUERRE POLYNOMIALS

POH-AUN LEEU,*, SENG.HUAT ONGb't

ANd H. M. SRIVASTAVA"'T

^Faculty of Information Technology, Multimedia University,63t00 Cyberjaya,

Selangor, Malaysia; blnstitute of Mathematical Scimces, tlniversity of Malaya,

50603 l{uala Lumpur, Malaysia; cDepartment of Mathematiis and Statistics,

University of Victoria, Yictoria, British Columbia Y|W 3P4, Canada

( Recefued 4 lanuary 2000; In final form 29 September 2000

)

The evaluation of an integral of the product of Laguerre polynomials was discussed recently in

this Jownal by Mavromatis [12] (1990) and Lee [9] (1997) [see also Ong and ke [1a] (2000)1.

The main object of the present sequel to these earlier works is to consider a family of such

integrals of the products of Laguerre, Hermite, and other classical orthogonal polynomials in a

systernatic and unified manner. Relevant connections of some of these integral formulas with

various known integrals, as well as the computational and numerical aspects of the results

presented here, are also pointed out.

I{eywords and Phrases: Laguerre polynomials; Hermite polynomials; Classical orthogonal

polynomials; Generalized hypergeometric functions; Lauricella functions; Appell functions;

Kampe de F6riet functions; Hypergeometric polynomials; Laplace transforms; Jacobi

polynomials; Pfaff-Saalschiitz theorem; Gegenbauer (or ultraspherical) polynomials; Poly-

nomial expansions; (Clebsch-Gordan) linearization formulas; Recurrence relations

$2000 Mathematics Subject Classifications: Primary: 33C?-0,33C45; Secondary: 44410

; C.R. Category: G 1.4

*e-mail: palee@mmu.edu.my

re-mail: ong@omega.math.um.edu.my

rCorresponding author. e-mail: harimsri@math.uvic.ca

P.-A.LEE et al.

1. INTRODUCTION

The classical Laguerre polynomials *)@), of order o and degree n ir x,

given explicitly by

rP (*) : (" *n"),r r?n;a * l ; x) : ff rrr

(- n, -, - n; - ;- * ),

(t)

are orthogonal over the semi-infinite interval (0, m) with respect to the

weight function xoe-'; in fact, we have its well-known orthogonality

property (cf., e.9., Szegd l24l)z

fo* *""-,Lg)1,1rf,)6ya* : (.:").,, + t)6m,n

(ft(a)>- 1; m,n€N6:: NU{0}; N ::{1,2,3,...}),

where 6*,, is the Kronecker delta. Here oFo denotes, as usual, a generalized

hypergeometric function with p numerator and g denominator parameters.

Just as the other members of the family of classical orthogonal

polynomials (e.g., the Jacobi polynomials plffi)@), the Hermite polyno-

mials II"(x), the Gegenbauer (or ultraspherical) polynomials Ci@), the

Legendre (or spherical) polynomial Pn(x), and the Chebyshev polynomials

T,(x) and U,(x) of the first and second kinds), the Laguerre polynomials

occur in many different fields of investigation in the mathematical, physical,

statistical, engineering, and numerical sciences. Some typical examples are

found in quantum mechanics [2], communication theory [1], and numerical

inversion of the Laplace transform (cf.ll0, pp. 249-2511; see also [16]). In

some of these and other areas of applications, one often requires explicit

(closed-form) evaluations of integrals involving the Laguerre polynomials.

Motivated essentially by its need in the investigation of some problems in

quantum mechanics, Mavromatis [2] evaluated the following integral

involving the product of two Laguerre polynomials:

.tFz(-m,pl-1,F- 0+l;a*1,p,- B-n+ l;l) (3)

(S(p) > - 1; rz,n€ N6),

(2)

I*,n(p;a, B) :: lo* *r"*Lp) 1x1lf) 14ax

: (* :") (" * u ;r-' )',, *,,

INTEGRALS OF POLYNOMIAL PRODUCTS

where (and elsewhere in this paper) it is tacitly assumed that the

parameters involved are so constrained that no zeros appear

denominator on the right-hand side.

Subsequently, Lee [9] deduced the integral formula (3) as an easy

consequence when

r:2, p:F*1, and o:\t:42:1

of the following general result which was discussed, almost sixty-five years

ago, by Mayr [13] and Erd6lyi [4] (see also Srivastava and Manocha 123,

p. 260, Problem 2(ii)l):

various

in the

,Y'f,,-*tt...t-nrio1 * 1, ...ter+ l;), . ,+] (4)

(m(p)>0; ffi(o)>0; n7€Ns; i :1,...,r),

fo* *o-t "-,*z[1,)(rrx) . . . y@) e,,x)* : (", :r*) . . . (" :,*)Y

where d) denotes the first one of the four Lauricella's hypergeometric

functions of r variables defined by (cf., e.g. l2l, p. 331)

4\lo,br,. ..,b,i ctt... t c,i 21,..., zrl

:: $ (o)*'*"'*r'(b')o' "' (b')o' 4' ...+

r,,1:,ffih6 (s)

(lrtl + .. . -t lz,l < t),

in which we have made use of the Pochhammer symbol (or the shifted

factorial) (.\)" given, in terms of Gamma functions, bV (lL::l()+n)/I()).

A probabilistic derivation of the integral formula (4), which is based mainly

upon the moments of certain non-central Gamma distributions, is contained

in the work of Ong and Lee [14].

In the present sequel to the aforementioned works of Mavromatis [2],

ke [9] and Ong and Lee [14], the first two of which appeared fairly

recently it this Journal, our main object here is to consider a family of

such integrals as in (3) and (4) involving products of Laguerre, Hermite,

and other classical orthogonal polynomials. We also point out relevant

connections of some of these integral formulas with various known

integrals, as well as the computational and numerical aspects of the results

presented here.

P.-A.LEE et al.

2. A I'AMILY OF INTEGRALS

We begin by recalling the fact that Erd6lyi [4] actually gave a more general

result than the integral formula (4) in the form:

to* xo-re-"*f lbr;c1;)1x) . . .tr1(b,;c,;\,x)dx

:ffrytlo,tr,...,b,ict,...,r,,+,,*] (6)

(ft(p)>0; W(")>ft(Ar + "' +l')),

which, in view of one of the hypergeometric representations given by (l),

yields the special case (4) when

bj:-nj and ci:oi*l (n7eNs; i:1,...,r).

Indeed the general integral formula (6) can be found reproduced in many

subsequent works (see, for example, Erd6lyi et al. 15, Vol. I, p. 216, Entry

4.22(14)l and Srivastava et al. 122, p. 49, Eq. II.7(6)1, 123, p. 260, Problem

2(i)1, and 121,p.285,Eq.9.4 (35)l). See also a recent application of (6) in the

derivation of some general finite multivariable hypergeometric summation

formulas given by Padmanabham and Srivastava [5].

Next we set .\,: o in (4) and apply the familiar Chu-Vandermonde

theorem I2l, p. 19, Eq. 1.2 (21)1. We thus find from (5) that

fo* *o-t"-" r!{.1,"'t^, x)\ rf')(ox)d*

:,.[ { ('' ;"')} (" . ::-')y

where the multivariable hypergeometric function occurring on the right-

hand side is a (Srivastava-Daoust) generalized Lauricella function of r-l

variables )r,...,L (r:2,...,r)

oo

(see, for details, l2l, p.38, Eq. 1.4(24) et seq.l).

I prp-or, -ryi..'; -nr-tl I

I )r ),-r I

.r?;l;::;ll i,. ,'+ I (7)

lp-or-rr, ar*l;"'; or-r*l; l

(ft(p)>0; ft(o)>0; nr€N6; i :1,...,r),

INTEGRALS OF POLYNOMIAL PRODUCTS

In its special case when r:2,the integral formula (7) immediately yields

lo* *-, "- "* rg) 1xx1 ilo)

1o x) dx : (* ;" ) (" . u,-, ) Y

/ n'^'t1^ D '\\ (8)

.rfr( -frt,ptp- 0;a+l,p- p-";;)

(m(p) > 0; ffi(o) > 0; rn,ne N6).

The integral formula (8) is a known result recorded, for example, by

Prudnikov et al. ll7 , p. 478, E;ntry 2.19.14.81, who also give its further special

case [17, p.478,Bntry 2.19.14.15] when

p:lt+1 and A:o:1,

which is precisely the integral formula (3). As a matter o[ fact, this special

integral formula (3) itself was proven, almost three decades before

Mavromatis ll2l, by Carlitz 13, p. 339, Equation (17)1, who also deduced

each of the following consequences of (3):

which obviously leads us to the orthogonality property (2) when 0: a;

(1c)

which was proven earlier, in a markedly different way, by Feldheim 16, p.2(1,

Eq. (1.12)1, but was stated there without the factor l@+A+ 1). It is thl,s

error in Feldheim's formula (10) that seems to have been carried over in

Erd6lyi et al. 15, Vol. II, p. 293, Entry 16.6(4)1, but (fortunately) not ia

Gradshteyn and Ryzhik 17, p. 845, Entry 7 .414.81.

Carlitz's formula (3) was applied by Srivastava [19, p. 211] with a view to

evaluating a certain integral involving the product of two Bessel pol1-

nomials y"(x;a,b).

Next, by applying a familiar generating function 123, p. 84. Equation

1.ll(14)l in conjunction with the multinomial expansion [18, p. 329t,

Eq. 9.aQ20)1, we can easily derive the following alternative form of the

/a*m\ /B-a*n-m -l\

I-,,(a;a,p): I l{ ' lr(a + 1) (9)

\ m /\ n-m /

($l(o) > - 1; lz,n€ Nsl n2m),

r^,n(a t 0;o, g): (-r)-*, (":*) (u :" ).r, * g + ry

(W(o+ A)>-l; n,n€Ns),

308 p.-A.LEE et al.

general integral formula (4):

[* *o-, " o, rf) etx) . . . y@) 1x,x)dx

Jo : o-pr(p* nt + . * ",)il{rY}

ag' [ - et - ft;t..., -dr - nr,-k1,...,-nri

l-p-nt- ...-n,,!. gl

rr:;'''ti (ll)

(m(p)>0; ffi(a)>0; n,€N6; j:1,...,r),

where rf;) denotes the second one of the four Lauricella's hypergeometric

functions of r variables (cf., e.g.,l2l, p.33, Eq. 1.4 (2))).

By setting l,: o in (l l) and appealing once again to the aforementioned

chu-vandermonde theorem, we obtain the following arternatiye form of the

integral formula (7):

f

o* *o-'

"-o'E {.rr',^, .l} LLi'' t' -) d x : o- e t (p+ N,-, )ii { r#}

. (n,+o,- p-N, r\,r,2,..,: [ ' - P*o'*t''-N'- 1:

\ n, )r z'o'"''o | .

Ll - p- N,_t, I - p*a,-N,_ r :

-Q.1 -nlr-nli,..i -Ar_t-tlr_1t-ttr l) I

? ,....o | ,t'

At ')', I (12)

;J

(n(p) > 0; ft(o) > 0; z7 € Ns; j:1,...,r),

where, for convenien@, N, ::D=tnj.

For r:2, the integral formula (12) reduces at once to the form:

lo* *o't "-,, $t 1x1$t 1ox) dx

: ("-*:u- e)t(e+m) -'(-:)^

. rFr(- ffi,-e - m,l - p+ p + n - m;l - p - m,l - e+ p - m;1)

(D(p) > 0; ffi(o) > 0; rn,z€ Ns).

( 13)

INTEGRALS OF POLYNOMIAL PRODUCTS

The integral formulas (11), (12) and (13) can indeed be deduced also

from (4), (7) and (8), respectively, by reversing the order of terms of the

finite hypergeometric series occurring on the right-hand sides.

Yet another interesting expression for the integrals in (8) and (13)

can be found by appealing directly to the known generating function

V3, p.84, Eq. 1.11 (14)l and the case r:2 of the multinomial expan-

sion [2], p. 329, F,q. 9.4 (220)), each of which was applied in our

derivation of the integral formula (11) above. We thus obtain the integrzrl

formula:

-mrPi-i -n; ,\

, -;,

-a-mt-i p-0-n;

(S(p) > 0; ffi(o) > 0; z, z € N6)

in terms of Kamp6 de Fdriet's function (cf. [2], p. 27, Eq. 1.3(28) et seq.7),

which is the case r:2 of the (Srivastava-Daoust) generalized Lauricella

function occurring in (7) and (12).

In its special case when p:p+l and o:), the integral formula (14)

reduces obviously to the following symmetrical form lcf. Eq. (3)l:

/ m I a - p"- l\ /r+0-p- I \-.

I^,(p'.".0):I m ,) (' n )rfu+t)

. tFz(-m,-n,Fl1;p-a-m*l,p- A -n*1;1) (15)

(W(p) > - l; m, zt € N6),

which, for p:o-1 or tt:A- 1, yields thefurther special cases:

to* *-"-*ra)6xyrf)loxro*: (*.;- ,)(". u,- ,)Y

r^,,(d - t;o, g): (-1)- (u:r").,", (16)

r*,,(g - r;a,0): (-1)' (":r*)r,r,, (17)

and

where we have also applied the aforementioned Chu-Vandermonde

theorem.

(14)

:l

.?fl l,

which (for

erty (2).

310 P-A.LEE et al.

The integral formulas (15) and (17) were given earlier by Carlilz

[3, p. 340]; only the special case (17) was recorded by Prudnikov et al.

ll7, p. 479, Entry 2.19.14.171. The following combinatorial series repre-

sentation for the integral l*,,(p;o, B) was given by Rassias and Srivastava

[18, p. 171, Eq. (19)]:

r*,,0,; q, e) : tr)^*^ri,*,) X ("-_"r) (: _i) (r.oo) (r8)

(S(p) > - 1; z,z€ N6),

p:a:g) leads us immediately to the orthogonality prop-

3. FURTHER INTEGRAL FORMULAS

The classical Hermite polynomials H,(x) are given explicitly by

lnlzl /

H,(x):[f-,1- (;r)+(2*)" 'o : (2x)n2F() lo,r,-,,,-'-11],

(1e)

where, and in what follows, we find it to be convenient to denote the

N-parameter array: )+j-l

simply by A(/V;,\), the array being empty when N:0.

By means of a familiar technique based upon generating functions,

which incidentally was applied in our derivation of the integral formula

(11), Carlitz [3, pp. 338-339, Section 3] also evaluated the integral:

t*,n(p;a,B) :: fo* o***" exp(*)n^@)H,(x)dx (20)

when (i) ft(p)> -l and m,neNs; and (ii) p: -1, m,n€N6, and

m+neN. In view of the hypergeometric representations in (1) and (19),

we consider the generalized hypergeometric polynomials (y' Brafman [2];

INTEGRALS OF POLYNOMIAL PRODUCTS 311

see also Srivastava and Manochal23, p.136, Eq. 2.6(l)l):

4 lor, . . ., apl bt, . . ., b, : xf :: 7,1a0 rq [A(N; -n), ar, . . ., ap', b1, . . ., bo; xf

(Ne N; r,p,{€ Ne),

(21)

so that, obviously,

B)l- ;a* r : xl : (' ;" ) 'r1"r1,1 and

,,2l ll /1-.\-nu/..\ (22)

Bil-,-, -F | : (2x)-'H,(x).

In terms of the multivariable (Srivastava-Daoust) generalized Lauricella

function occurring in (7) and (12), we find from the definition (21) that

lo* *n"**fi{si;[,,1r, "' ,rtjpt; bir, " ' ,biq' t l1x)]ax

|(9) .t,N,+p,;. ;v,ap,

- op ' o, qti.; q,

I pt A(Nl ;-r1), arr,...,arpri"'l L,(N,;-n,), ail,...1arp,i I

I r Il

I d""'o I

[-: bn,...,brq,i...i bt,...,b,q,i ]

(n(p) > 0; E(o) > 0; N; € N; ni,pi,4i e Noi j : 1,...,r). (2?)

For r:2, the general integral formula (23) readily yields the following

unification of the above-discussed numerous integrals of products of

Laguerre, Hermite, and other polynomials:

[* *o-t r-o' BXlrr, . . ., api bl, . . ., bn : )x) BI[rr, . . ., c,', d;, . . ., d, : pxldx

Jo l(P) d,u+p;N+,

- ,o ' lt l, r

f p, A'(M;-m), a1,...1apt A(N; -n), cl,...,cr1

I

I

I

f-: b1,...,bq; dr,...,dr;

(n(p)>0; W(o)>0; M,Ne N; ffi,n,p,4,r,s€Ns)

)u]

(24)

in terms of the Kamp6 de Fdriet function occurring in the integral

formula (14).

312 P.-A. LEE et al.

4. ADDITIONAL CONSEQUENCES AND APPLICATIONS

In terms of the Appell function r'2 which is simply ff) defined by (5), a

special case of the integral formula (4) when r:2 yields

r9,l,r) (p; a, B) :: fo* *o-,

" "*y@) 1xx)Lf) (p,x)dx

:(^+0\f,?+0\r(p)

:\ m i \ , ) "'

'rrlp,_-m1-ni(t - 1,0* r, l,4l (25)

L o o)

(m(p) > 0; ft(") > 0; m,n 6 No),

so that, by comparing the definitions in (3) and (25), we have

I*,0r;o,g) =t*:),J)(tr-t l;a,p). (26)

By applying a known reduction identity [21, p. 305, Eq. 9.a(108)], we easily

find from the integral formula (25) that (cf. ll7, p. 477, Enlry 2.19.14.61)

rSi,d 1ar r;o, "l : ( * ;") (' :") #*+ @ - \)*(o - 1,.)"

' ,, l' ,1 pr1

' ,r' |. - nl, -ni" + t; ,o _ 41o _ pyJ

(ft(o) > - l; ft(") > 0; m,ne No),

which, in view of the polynomial identity:

2F1(a,b;c;z) :tr$#+ 2F1(a,b;a* b - c * t;t - z) (28)

(a or beZ[;ceZ;;Zo :: {0,-1,-2,...}),

assumes the equivalenl form (cf.l5,Vo1. I, p. 175. Entry 4.11 (35)l; see also

17, p. 844, Entry 7.414.41 and [7, p. 477, Entry 2.19.14.6]):

t$,;),D 1a t | ; a, *1 :19ffi]J!#9#

.rF,l -m,-ni-m- o(o-\-/') I

L n-"tffi1 (2e)

(ft(o)> - 1; ft(")> 0; m,n6 No).

INTEGRALS OF POLYNOMIAL PRODUCTS 3I3

Next, since [2], p. 318, Eq. 9.4(163)]

F2lc * C - l,b,b;c,c';x,yl : Falc * y' - 1,b;c,c';x(1 -y),y(l - x)], (30)

where Fa denotes the Appell function of the fourth kind [2], p.23,Eq.1.3

(5)1, yet another special case of (25) when p: a+ {3+l and m: n yields the

integral formula:

rf,;^,d 1a + o + t;a, o) : (":") (" :u)\##:I

I

r,+1,+ B+1,-n;a*t,g+ r;l (r -:),:(,-))l (3r)

(W(a+0)>-l; ft(")>0; re N6).

Upon setting \:p in (31) and applying another known reduction identity

l2l, p. 314, Eq. 9.4(149)1, we obtain

tf,;^ \ 1a + o + | ; a, p) : (" :") (, ; o)t (;.j=g,i t)

,tr[- ,,lO+0+r,]t,+ a+g;a*t,p*t,?(, -])] e2)

(ft(a+0)>-r; E(")>0; ne Ns),

which, in the further special case when o - B, reduces immediately to the

form:

tf,;^,\(zr,1 l;o,o)

: (":")'\P*!,r,1-n,o+l;o * r'? (, -])] (33)

/1\

(of"l ,-ri W(")>0; re No/

or, equivalently,

,tor\.x)12a + llo,o)

_22"r(d+ (tlV))t(n+ (tl2)) r(a * n1_ t)

r(nt)2 62o+t

,., [- n,o +],L-,,(, - ?)'] Q4)

(*,", , --, E(o)>o; ,e ruo),

which follows from (33) by means of the polynomial identity (28).

314 P.-A.LEE et al.

For ,l:l (and ov---+p\, the integral formula (34) would provide

the corrected version of a known result. (cf [5, Vol. l, p. 174, Entry

a.11(30)l; see also 17, p.845,Fntry 7.414.101). Moreover, in view of the

aforementioned Chu-Vandermonde theorem, it is not difficult to deduce

from (34) that

tlr;t'r) 1za* I ; a, a) = Ir,n(2a; a, a)

_ z2"l(a + (t l2)) {r(a * n + t)}2 (3s)

- 1nt12r/ir1o + l) t^'

(o(,), ' \

\ '-'i reNo)'

which indeed is an obvious special case of the familiar result (10) when

m:n and a:0.

In view of a known generating function 123, p. 109, Eq. 2.3 (20)) for the

classical Jacobi polynomials pf'a) (*) of indices a and B (and of degree n

in x), defined by

P@,p, @),: x (,:7) (' ; r) (+)' " (=)-

/n+o\ r l-x\

:( n )'o'(-''"+o+n-l;o*t'i)' (36)

the integral formula (31) can easily be rewritten in the form:

tf;^,d1a+gtt;a,A)

_ r(o +n + l).\f, +0\ r(a+B+t + r)

nlo"-B+t fu\"-t ) f(of k+ l)

(*)r,tt',('^r(;y;:)") r,,r

(ft(a+ 0)> -r; ft(")>0; neN6).

This last integral formula (37) is obviously not convenient to use in the

case when ): p. With a view to overcoming this difficulty, we first rewrite

(25) in the form:

79,:)'D(p;", P)

_ (m+a\(n+B-p\r(p)

-\ m i \ n )"'

INTEGRALS OF POLYNOMIAL PRODUCTS

(mk) > 0;

-m,p- P; -n;

o*l; -;

rz, r € Ng), (38)

where we have made use of the polynomial identity (28). Now we shall make

use of a (presumably new)hyperyeometric transformation analogous to (30)

above:

,,r l

-;-;

c; c';

which holds true, by the principle of analytic continuation, whenever each

double hypergeometric series converges absolutely. In its special case when

c+c':a+b-t 1, the hypergeometric transformation (39) reduces immedi-

ately to a well-known reduction formula l2l, p. 304, Eq. 9.4(100)l for

the Appell function -E4 occurring in (30) above. Furthermore, a limit case

of (39) when a --+ oo is precisely the known hypergeometric transformeL-

tion (30).

Our proof of the hypergeometric transformation (39) is based largely

upon Appell's formula [8, p. 304, Eq. 9.4(99)], and also upon the Pfaft-

Saalschiitz theorem 120, p.95, Problem 25(i)l; we choose to omit the detaiis

involved.

In its special case when a: -n (n e Ne), c: a* l, and b: C : 0+1, the

hypergeometric transformation (39) assumes the terminating form:

31s

I , -*l

o' ,)

*(r-y),ytr-,).l, (3s)

l

lc+C-l: a,b) a,b;

rl',1:? I

I a+b: q C;

lc+C-l,o,b:

-3:0:0 I

: f t,t;t I

I a+b:

x(l -y),y(l -r)],

(40)

pi

p-0-n:

W(o)> 0;

"ffi$ [

lr,+A+l:

rl.i.l I

lB-n+r,

f-n

: r?,?ll I

L

a*l; -;

,at13-t1,0*l:

-n,B+l; -n;

0-n*l: a*l; 0+l;

,, r'l

l

316 P.-A. LEE et al.

which is applicable to (38) with, of course, q: B and m:ry so that

rf,;t'il1r,l- B*t;a,a)

_ ( "+ "\ ( n - 0 - 1\ r(a+0+ l)

-\ " )\ " )---drd+t

f-n,o+p+1,0+1, I

ri;?;il #,('-) (, -#) I

I B-n*t: a*1; g+t; I

(n(o+0)>-l ft(")>0; ne Ns). (4r)

By means of a known result [23, p. 145, Eq. 2.6(31)], which provides a

generalization of the generating function used in our derivation of (37)

above, we can rewrite the integral formula (41) as follows:

tfl'il1a* B + t;a,a)

:(_l),(n+a\r(o+0+t1

- r-,r \ n ) -- n, t.-,

n("u- r) (r * i* u) (o r)' (* - r)-

Py(*##r) ro,r

(ft(a+0)>-r ft(")>0; ne N6)

in terms of the Jacobi polynomials defined by (36).

Since

rf,,,)1x1: (":")(' *,^) ,,+tt/z\(x), (43)

a further special case of this last integral formula (42) when s: B readily

yields

zfl,d1za11;a,o)

( n+ a\ r(2a + l)

: \ " )--F-

tr-,r* ( ; X' - ry)*r ri+['l

/z) (L#Wal

(o,",, -1, ft(,)>o; nervo)

(44)

INTEGRALS OF POLYNOMIAL PRODUCTS 317

in terms of the Gegenbauer (or ultraspherical) polynomials Ci@) of index z

(and of degree n in x).

The integral formula (44) provides the corrected version of a result which

was given earlier by Howell [8, p. 1089, Eq. @.2)]; it appears incorrectly also

in a number of subsequent works (cf., e.9., Erd6lyi et al. 15, Vol. I, p. 175,

Entry 4.11(36)l and Prudnikov et al. ll7, p. 478, Entry 2.19.14.7!).

Moreover, in the special case of (42) when 0:0, the finite series on the

right-hand side would reduce to its only nonzero term given by /c:n. This

special case of (42) when 0:0 is recorded by (among others) Gradshteyn

and Ryzhik 17, p. 845, Enffy 7.414.131.

Starting from a known generating function 123, p. ll2, Eq. 2.3(34)l and

setting *:; -_ll* P"!,xr and,: (r-lll)", (4s)

o(o-),-1.r,) \ a. /'

it is easily seen that

(r -,)-,-' try] -o-u-',r,[o1r,, + o+ 1); " + r, rffi1

: (, - n'F-(.;. u)(o;")-'

oP'gt ( o2 - (^ -

.k r-;o-ffta) [(,-*)l-

: i(-')'E(, '- r)(r * i* u)(*; ") '

o6.0, (; - (^ + 7r)a F 2)P\

P;"'\__ffi), (4()

so that the integral formula (42) can be rewritten in yet another form (c1.,

e.g., Gradshteyn and Ryzhik l7 , p. 845, Entry 7 .414.121and Prudnikov el a/.

}7, p. 477, Entry 2.19.14.5)):

tf;^,u) 1a -t B + t; a, a)

/n+a\f(o+B+l)

-\ , ) nl

#{rt- r)-o-r [o1,;1-"-r-rro, [o1r, a + 0 -rl);a * l; *#Gf] ]1.=.

(E(a+0)>-l; ft(")>O; neNe), (47)

318

where, for convenience,

P.-A.LEE et al.

(48)

The integral formulas (33) and (34) provide remarkably simplified versions

of (4Q with \: p.

5. COMPUTATIONAL AND NUMERICAL ASPECTS

Closed-form evaluations of the various integrals considered in this paper are

required in many areas of applications such as the ones pointed out in

Section 1, and also in several other diverse fields of physical, astrophysical,

and engineering sciences leading naturally to problems involving polynomial

expansions and Clebsch-Gordan type linearization relations associated with

the Laguerre and other families of orthogonal polynomials (see, for details,

[20]). These have been achieved in our paper by deriving explicit formulas

involving terminating series.

The computations of the terminating series occurring in our closed-form

evaluations are not difficult, except possibly for overflow problems in the

evaluations of the Gamma functions. It is well known that the Gamma

function l(x) overflows very quickly on a computer (for x > 120, depending

on machine precision). In Section 4 the integral formula (4) for the special

case of two Laguerre polynomials was given by Eq. (25) as an Appell

function.F2, which is of importance in various applications as pointed out in

Section 1. Consequently, the various special cases given in the forms of

terminating hypergeometric 2F1 series or finite sums of Jacobi (or, more

simply, Gegenbauer) polynomials are of great utility in computation. For

these cases, the problem of evaluation of a double sum in ,F2 has been

replaced by that of a single sum. In this connection, we note also that the

integral formula (44) has the simplified versions (33) and (34) in terms of

terminating hypergeometric 2F1 series.

As an illustration, we consider the computation of (37). In this example,

computations of most of the Gamma functions are avoided by employing

recurrence relations for the quantities computed.

We begin by rewriting the integral formula (37) as follows:

zf,l't')1a * g * t;a, fr): ,t^rrutc)rf,'a ('Tlp,1, @s)

k:0

O(r):- "+ry

INTEGRALS OF POLYNOMIAL PRODUCTS 319

where, for convenience,

r(n;k),:wfi\ (::t1H## (Y)- (x * ti

By considering the quotient r(n;k-tl)lr(n;k), the following recurrence

relation is obtained:

r(n;k*t :ffi(*),*,o (k : 1,...,n) (so)

with, of course,

r(n;o):W#(:u)H#

Now, in order to compute r(n;O), we may use

W,g;01

(n+1)(n+2)

(s1)

(s2)

(s3 )

r(r+ 1;0) :

with the initial value:

r(o:o) : l9;#+I

Thus, with a view to evaluating r(n;k), we first compute r(0;0) which

involves only one Gamma function l(a+0* 1). There are many efficient

programs to compute l(a+p+l) (see, for instance, Algorithm AS245

provided by Macleod [11]). For large arguments of the Gamma function

f(x), Stirling's approximation may be used. Next r(n;0) is evaluated by the

recurrence relation (52). With this numerical value of r(n;O), r(n;k) is then

calculated by appealing to the recurrence relation (50).

The Jacobi polynomials p|'B) (*) occurring in (49) satisfy the following

three-term recurrence relation (see, for example, p3, p. 7ll):

2(n + t)(a -t 0 * n+ l)(o + 0 + zflrftBr) 1x1

: (o * g + zn+ l) [(o + B + 2n)(a * g * 2n + 2)x * "' - frl rf,$ 1x1

- 2(a + n)(0 + n)(a + 0 + zn + zlrf,Brt 61 (z e N). (s4)

The recurrence relation (54) enables p9$ @) to be computed rather

quickly. It also obviates the need to calculate series given in the form of

hypergeometric z.F'r series.

P.-A.LEE et al.

Acknowledgments

Many parts of the present investigation were finalized during the third-

named author's visits to Multimedia University at Cyberjaya (Selangor) and

the University of Malaya at Kuala Lumpur in March 2000 and July 2000.

This work was supported, in part, by the Natural Sciences and Engineering

Research Council of Canada under Grant OGP0007353.

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