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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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