Some integral representations for the Jacobi and related hypergeometric polynomials

              Rev. Acad. Canar. Cienc., XIV (Núms. 1-2), 25-34 (2002)
SOME INTEGRAL REPRESENTATIONS FOR THE JACOBI AND
RELATED HYPERGEOMETRIC POLYNOMIALS
H.M. Srivastava
Department of Mathematics and Statistics
University of Victoria
Victoria, British Columbia V8 W 3P4
Ganada
E-Mail: harimsri@math.uvic.ca
Dedicated to Professor Nácere Hayek Calil
on the Occasion o/ his Eightieth Birthday
Abstract
The main object of this paper is to present sorne double- and triple-integral
representations for two general families of generalized hypergeometric poly­nomials.
lt is also shown how a certain triple-integral representation for an
obvious variant of the classical Jacobi polynomials, which was proven in this
Revista by A.K. Chongdar and N.K. Majumdar [Rev. Acad. Canaria Cienc.
13 (2001), 59-64], can be deduced fairly easily from the corresponding known
result for the Jacobi polynomials themselves.
2000 Mathematics Subject Classification. Primary 33C20, 33C45; Secondary 33C65.
Key words and phrases. Integral representations, Jacobi polynomials, Laguerre polynomials, Bessel poly­nomials,
Hermite polynomials, hypergeometric functions and polynomials, Laplace and inverse Laplace trans­forms,
Kampé de Fériet's function, multivariable hypergeometric polynomials.
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l. lntroduction, Definitions and Preliminaries
Let (.\),, denote the Pochhammer symbol (or the shifted factorial, since (l)n = n! for
n E No) defined (for .\, v E C and in terms of the Gamma function) by
(,\) ·= = (1.1)
r (.\ + v) { 1 (v =O;,\ E C \ {O})
V • r (,\) ,\ (,\ + 1) ... (,\ + n - 1) (v = n EN:= No \ {O}; ,\E C))
where N0 is the set of nonnegative integers. Also, as usual, we denote by pFq a generalized
hypergeometric function with p numerator and q denominator parameters.
The classical Jacobi polynomials P~a,/3) (x), oforder (a, {3) and degree nin x, defined (in
terms of the Gauss hypergeometric 2F1 function) by
p~a,/3) (x) := ( ª: n) 2F1 (-n, a+ f3+n+1; a+ 1; 1 ; x) (1.2)
or, equivalently, by the Rodrigues formula:
(-ir (1 - x) - 0< (1 + x) - {3
PJª·13) (x):
· D~ { (1 - x)°'+n (1 + xl+n}
are orthogonal over the interval ( -1, 1) with respect to the weight function:
w (x) := (1 - x)°' (1 + x)13 ;
in fact, we have ( cf, e.g., Szego [14])
[~ (1 - x)°' (1 + x) 13 p~a,/3) (x) p~a,{3) (x) dx
2a+f3+1 r (a + n + 1) r ({3 + n + 1)
n! (a + f3 + 2n + 1) r (a + f3 + n + 1) Óm,n
(a > -1; f3 > -1; m, n E N0 ) ,
where Óm,n denotes the Kronecker delta.
(1.3)
(1.4)
(1.5)
In recent years, a great deal of attention seems to have been paid to an obvious variant
of the classical Jacobi polynomials PJª·13) (x). These so-called extended Jacobi polynomials
FJª·13) (x; a, b, e), studied (among others) by Izuru Fujiwara (1928-1985) in an attempt to give
a unified presentation of the classical orthogonal polynomials (especially Jacobi, Laguerre,
and Hermite polynomials), are defined by the Rodrigues formula:
FJª·13l (x; a, b, e) := (-~t (x - a) -ª (b - x) - 13
n.
· D~ { (x - a)°'+n (b - x)f3+n} ( e:=_.\ >o) b-a
(1.6)
and are orthogonal over the interval (a, b) with respect to the weight function [ cf Equation
(1.4)]: .
w (x; a, b) := (x - a)ª (b - x)13 • (1.7)
The polynomials FJª•13) (x; a, b, e) are essentially those that were considered by Szego [14, p.
58], who showed (by means of a simple linear transformation) that these polynomials are
just a constant multiple of the classical Jacobi polynomials P~a,/3) (x ). In fact, by comparing
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the Rodrigues formulas (1.3) and (1.6), it is not difficult to rewrite Szego's observation [14,
p. 58, Equation (4.1.2)] in the form (ej., e.g., Srivastava and Manocha [12, p. 388, Problem
11]):
or, equivalently,
FJª·/Jl(x;a,b,c) = {c(a-b)t PJª·/Jl c~~ba) +1)
= {-c(a-b)}n pJfJ,a) (1- 2~x--bb)) (1.8)
PJcr ,fJ) (x) = {e (a - b)} -n FJcr,fJ) ( ~ {a+ b + (a - b) X}; a, b, e) . (1.9)
Thus, as already pointed out by Srivastava and Manocha [loe. cit.], the polynomials
FJcr,fJ) (x; a, b, e) may be looked upon as being equivalent to (and not as a generalization of)
the classical Jacobi polynomials PJª·fJ) (x ).
Furthermore, by recourse to certain limiting processes, it is easily seen that the polynomials
FJcr,fJ) (x; a, b, e) would give rise to the Laguerre and Hermite polynomials (and indeed also to
the Bessel polynomials) justas the classical Jacobi polynomials PJª·fJ) (x) do. Consequently,
the main purpose of Fujiwara's investigation [5] is already served by the classical Jacobi
polynomials themselves.
Even after the aforementioned observation by Szego [14] and others (ej., e.g., Srivastava
and Manocha [12]), the polynomials FJcr,fJ) (x; a, b, e) have been (and are still being) made,
in recent years, a tool for the purpose of generalizing what is already known in the context
of the classical Jacobi polynomials (see also Pittaluga et al. [8] and González et al. [6]). For
example, in a paper published very recently in this Revista, Chongdar and Majumdar [4]
derived a triple-integral representation for the product:
F(a,fJ) (x· a b e) F(a',fJ') (y· a b e)
m ' ' ' n ' ' '
with a view to "generalizing" the corresponding known result of Chatterjea [3] for the prod­uct:
Pj;:•fJ) (x) PJª',fJ') (y)
of two Jacobi polynomials. Since x and y are not the variables of integration in each of these
triple-integral representations, the equivalence of the Chatterjea [3] and Chongdar-Majumdar
[4] results can be demonstrated fairly readily by means of the relationships (1.8) and (1.9).
In this sequel to several earlier works on the subject (including, for example, [3] and [4]), we
aim at presenting sorne double- and triple-integral representations for two general families
of generalized hypergeometric polynomials. Relevant connections of the results considered
here with those obtained in earlier works are also indicated.
2. The Main Integral Representations
For the product:
L~) (x) L~ª) (x)
of two Laguerre polynomials defined by
L(a) ( ) ·= ~ (n + ª) (-xl
n X · L.J n - k k!
k=O
(2.1)
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or, equivalently, by
L~ª) (x) := ( ª: n) 1Fi(-n; a+ l; x) (2.2)
in terms of the confluent (Kummer) hypergeometric 1F1 function, Watson [15, p. 207] gave
an elegant integral representation which was subsequently generalized by Carlitz [2] in the
following form:
L(o:) (x) L(fJ) ( ) = 2a+fJ+m+n f (a+ m + 1) r (/3 + n + 1)
m n y 7f2 r ( Q + /3 + m + n + 1)
! 1T/2 !1T/2
. J~',fl (19, cp) L~::i (n [x, y; 19, cp]) dcp di?
- 7T/2 - 7T/2
(2.3)
( m, n E No; a + /3 > -1) ,
where, for convenience,
J!..~I!) (19, cp) := e{(m- n)<pi+(o:-fJ)t9i} cosm+n cpcosª+fJ 19 (2.4)
and
n [x, y; 19, cp] := cos 19 [xe(t9-rp)i + ye- (t9 - rp)i] . (2.5)
coscp
Integral representations for similar products of many other polynomials have since then
appeared in the mathematical literature. A detailed account of these results may be found
in a paper by Srivastava and Joshi ( cf. [10, p. 920]; see also [9]), who also gave a general
(p + q + 2)-dimensional integral representation for the product:
[
(ap),a; l [ (a~),a'; l <I>m X '1>n Y ,
(bq) + 1; (~) + 1;
where, and throughout this paper, (ap) abbreviates the array of p parameters
a1, ... , ar>,
with similar interpretations for (bq), et cetera, and
[
(ap),a; ] ·- (a)n ¡-n,a+n,a1, ... ,ap; l <I>n X .- - 1 -
p+2Fq X
(bq); n. b1, ... ,bq;
(2.6)
in terms of an obviously terminating generalized hypergeometric series.
The family of hypergeometric polynomials defined by (2.6) possesses a generating function
in the form:
oo [ (ap);a; l n_ - <> [ ~a,~a+~,a1, ... ,ap; _ 4xt l L <I>n X t - (1 - t) p+2Fq . (l _ t)2
n=O (bq); b1, ... ,bq,
(2.7)
(ltl < 1)
and includes, as a special or limit case, severa! known polynomial systems such as the Jacobi
polynomials p~<>,fJ) (x) defined by (1.2), the Bessel polynomials of Krall and Frink [7]:
Yn (x; a, (3) := 2Fo ( -n, a+ n - 1;-; -~) , (2.8)
the general Rice polynomials (cf., e.g., [12, p. 140, Equation 2.6 (13)]):
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HÁª·13l ((,u, v) := e~: n) 3F2 (-n, a+ /1+n+1; (;a+ 1, u; v), (2.9)
and the Laguerre polynomials L~ª) (x) defined by (2.2), since
. ¡-n,a+n,(a,,);x]- ¡-n,(a,,); l hm 11+2Fq - - 11+1Fq x .
a-too (bq); a (bq);
(2.10)
Now, starting from Carlitz's formula (2.3) and making repeated use of the relationship:
p(a,/3) (x) = 1 f 1 (1og ~) a+f3+n L(a) (~ (1 - X) log ~) dt
n f(a+/1+n+l)}0 t n 2 t
(2.11)
(a+/1>-1; nENo),
which is, in fact, equivalent to Feldheim's formula [12, p. 94, Problem 24]:
p(a,/3) (x) = l f 00 tª+f3+n e-t L(a) (~ (1- x) t) dt (2.12)
n f(a+/1+n+l)}0 n 2
(a+ /1 > -1; n E No),
it is not difficult to observe, by the principie of multidimensional mathematical induction
based u pon the Laplace and the inverse Laplace transforms, that ( cf [13, p. 424, Equation
(30)])
[
(¡u): -m,>.+m,(a,,);-n,μ+n,(c,.); l Fu:p+2;r+2
v:q+l;s+l X, Y
(óv) : a+ 1, (bq); /1+1, (d,);
= !:!,_(a,/3) 11í/211í/2 ¡(a,/3) ({) rn)
m,n m,n 'r
-7í/2 -7í/2
. pu+l:p+l;r+l
v+l: q; s
[
-m - n, bu) : >. + m, (a,,);μ+ n, (e,.);
a+ /1+1, (óv) : (bq); (d,);
((x;D,,,),"(y;D,,,)] d<p dD (2.13)
(>.>O; μ>O; a+ /1 > -1; m,n,p,q,r,s,u,v E No),
where J~f!l (fJ, cp) is given by (2.4),
and
2a+/3+m+n m! n! r (a+ 1) r (/1+1)
!:!,. (a,/3) ·= --..,--
m,n . 7f2 ( m + n) ! r (a + /1 + 1)
cos {) (iJ ) .
~(x;fJ,cp):=x--e -ip•
cos cp
and
cos {) (iJ )i r¡ (y;{), cp) :=Y -- e- -ip ' cos cp
F:~:;~ (p, q, r, s, u, v E No)
29
(2.14)
(2.15)
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denotes a general (Kampé de Fériet's) double hypergeometric function defined by (cf, e.g.,
[11, p. 27, Equation 1.3 (28)]; see also [1, p. 150])
[
a¡, . .. ,ap:a¡, ... ,ar; C¡, ... ,c,,; l pi;;:;~ x,y
{3¡, . .. ,{Jq:b¡, . . . ,b.; d¡, ... ,dv;
where, for convergence of the double hypergeometric series,
and p +u ~ q + v + 1,
with equality only when
{
lxl1/(p- q) + IYl1/(p- q) < 1 (p > q)
max {lxl, IYI} < 1 (p ~ q) ·
In light of the limit formula (2.10), the double-integral representation (2.13) with
would readily yield
X
X t-+ ~ '
y
y t-+ - ,
μ
and min {A, μ} -t oo
[
bu): - m, (ap); - n, (e,.); l vu:p+l;r+l
r v:q+l;s+l X, Y
(óv) : O!+ 1, (bq); {3 + 1, (d8 );
1
7r/217r/2
= Ó. (a,fJ) ¡ (a,{J) ({) rri)
m,n m,n 'Y - 7r/2 - ¡r/2
(2.16)
·F :!~~:;: [ f. (x; {), cp) , r¡ (y;{), cp) dcp d{)
-m - n, bu): (ap); (e,.); l
O!+ {J + 1, (óv) : (bq); (d,);
(a+ f3 > -1; m, n,p, q, r, s, u, v E N0 ).
Furthermore, since [11, p. 28, Equation 1.3 (30)]
[
O!¡, ... ' O!p : -; - ; l p:O;O
q:O;O X , Y
{3¡, ... '{Jq: - ;-;
[
O!¡, ... ,ap; l = vFq x+y ,
f31, ... '{Jq;
30
(2.17)
(2.18)
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a special case of (2.17) when
p=q=r=s=u=v=O
is precisely Carlitz's formula (2.3) in light of the definition (2.2).
Next, by the definition (2.16) in conjunction with the Eulerian Beta-function formula:
r (a) r (/3) = B (a, /3) := ¡1 t°'- 1 (1 - t)/J-l dt
r(a+/3) lo (a>O; /3>0),
we find that
[
-m-n,('yu): .A+m;μ+n; l F u+l:l;l
v+l:O;O X, Y
Q + /3+1, (8,,): --;--;
u
k+E +n ( -m - n) k+I lJ-1 ('Y i) k+I r( .A + m + k) r (μ + n + l) xk '![_
- ( + /3 + 1) n" (1' ·) r (.A+ m) r (μ + n) k! l! k .1-0 Q k+I u3 k+l
j = l
= r (.A+μ+ m + n) r1 t'Hm-1 (1 - tt+n-1
r(.A + m) r (μ + n) lo
[
-m-n,.A+μ+m+n,(1'u) :-;-; · rr .mv++l2:O:0;;O0 xt, y ( 1 - t ) ld t
a+/3+1,(8,,) :-;-;
= r (.A+μ+ m + n) r1 t'Hm- 1 (l _ t)¡.i+n- 1
r (.A+ m) r (μ + n) lo .
[
-m-n,.A+μ+m+n,('y,,); l · u+2Fv+l xt +y (l - t) dt
Q + f3 + 1, (8,,);
(.A> O; μ>O; m,n,u,v E No),
(2.19)
(2.20)
by means of the reduction formula (2.18). U pon substituting from (2.20) into the integrand
of (2.13) with, of course,
p = q = r = s =O,
we thus obtain the following general family of triple-integral representations:
[
(1'u): -m,.A+m;-n,μ+n; l r_,,·;2ú·2 x,y
(8,,): a+l; /3+1;
= r (.A+μ+ m + n) f1(<>,fJ) ¡1 ('/2 ('/2 e+m-1 (1 - tt+n-1 ¡C<>,/JJ (19 r.p)
r(.A + m) r (μ + n) m,n lo }_7r/2 }_7r/2 m,n )
[
-m-n,.A+μ+m+n,('yu); l u+2F,,+i n [xt, y (l - t); 19, rp] dr.p d19 dt
a+/3+1,(8,,);
(2.21)
(.A> O; μ>O; a+ /3 > - 1; m, n, u, v E N0 ),
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where ¡t:,;!!l ('19, <p) and f}.~;f!l are given by (2.4) and (2.14), respectively, and
n [x, y; '19, <p] = ~ (x; '19, <p) + r¡ (y; t9, <p)
is defined, as before, by (2.5).
3. Applications to the Jacobi and Related Polynomials
(2.22)
For u = v = O, each of the two-variable hypergeometric polynomials occurring on the
left-hand sides of (2.13), (2.17), and (2.21) would obviously reduce to the product of two
one-variable hypergeometric polynomials of the types involved in (2.6) and (2.10). Thus,
by specializing in this manner, we can deduce the corresponding double- and triple-integral
representations for the products of two such one-variable hypergeometric polynomials. In
particular, upon putting u= v = O in (2.21), we obtain
2F1 ( -m, ,\ + m; a + 1; x) 2F1 ( -n, μ + n; /3 + 1; y)
= r (..\ + μ + m + n) f}.(a,{3) { 1 !1f/2 !1f/2 t'Hm-1 (l _ t)"+n-1 ¡(a,{3) ('19 <p)
r (,\ + m) r (μ + n) m,n lo - 1f/2 - 1f/2 m,n ,
· 2F1 (-m - n, >. + μ + m + n; a+ /3 + 1; n [xt, y (1 - t); t9, <p]) d<p d'l9 dt (3.1)
(..\>O; μ>O; a+ /3 > -1; m, n E N0 ),
where ¡t:,;!!l (t9, cp), n [x, y; '19, cp], and f}.~;f!l are given by (2.4), (2.5), and (2.14), respectively.
Finally, by setting
1-x 1-y f3 r--+ ¡, ,\=a+ f3 + 1, μ = ¡ + ó + 1, x r--+ - 2-, and y r--+ - 2- ,
and applying the definition (1.2), it is easily seen from (3.1) that
p(a,{3) (x) p('Y.ó) (y)_ 2°+7+m+n r (a+ m + 1) r (7+n+1) r (a+ /3 +'Y+ ó + m + n + 2)
m n 1í2 r( a + /3 + m + 1) r (r + ó + n + 1) r (a + 'Y + m + n + 1)
. [1 !1f/2 !1f/2 ta+fJ+m (1 - t)"f+ó+n J!::_:;¿J (t9, <p)
lo - 1f/2 - 7f/2
· P~ª++;.7·/3+ó+1l (1 - O [(1 - x) t, (1 - y) (1 - t); t9, cp]) d<p d'l9 dt
(3.2)
(a+/3>-1; 1+ó>-l; a+7>-l; m,nENo) ,
where ¡t:,;..7l (t9, <p) and O [x, y; '19, <p] are defined, as before, by (2.4) and (2.5), respectively.
The triple-integral representation (3.2) provides the corrected (and notationally slightly
modified) version of the main result of Chatterjea [3, p. 756, Equation (2.13)], justas it was
observed also by Chongdar and Majumdar [4, p. 63, Equation (2.9)]. More importantly, in
view of the relationships in (1.8), the main result of Chongdar and Majumdar [4, p. 62,
Equation (2.8)] can be deduced from Chatterjea's result (3.2) itself by first setting
2~-aj 2~-aj
x r--+ a _ b + 1 and y r--+ a _ b + 1
or, alternatively,
1 2 (x - b)
X f--t - ~-~
a-b
and
2 (y - b)
yr--+1 - a- b '
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and then making sorne obvious notational adjustments. Thus, in the latter case, the argu­ment
z of the Jacobi polynomial of the integrand of the triple-integral representation (3.2),
in the F-notation of (1.8) and (1.9), is given by
a+ b - z = ~ {a+ b +(a - b) (i - a: b n [(x - b) t, (y - b) (i -t); 19, cp])}
=a - n [(x - b) t, (y - b) (i - t); 19, cp]' (3.3)
that is, by
z = b + n [(x - b) t, (y - b) (i - t); 19, cp]
= b + { (x - b) te(t'J-rp)i +(y - b) (i - t) e - (t'J-rp)i} cosl9, (3.4)
cos 'P
in view of the definition (2.5) and the relationship:
FJa,,B) (a+ b - x; a, b, e)= (-ir FJ.B,a) (x; a, b, e), (3.5)
which does indeed follow readily from the well-known relationship [i4, p. 59, Equation
(4.1.3)]:
P᪷.B) (-x) =(-ir p~.B,a) (x). (3.6)
Equation (3.4) does correspond to the argument of the Jacobi polynomial of the integrand
of the main result of Chongdar and Majumdar [4, p. 62, Equation (2.8)], which is essentially
the same as Chatterjea's result (3.2) simply rewritten (or translated) in the F-notation.
Acknowledgements
The present investigation was supported, in part, by the Natural Sciences and Engineering
Research Council of Ganada under Grant OGP0007353.
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