Integral representation for the product of two extended Jacobi polynomials

              
Rev. Acad. Canar. Cienc., XIII (Núms. 1-2-3), 59-64 (2001) (publicado en Julio de 2002)
INTEGRAL REPRESENTA TION FOR THE PRODUCT OF TWO
EXTENDED JACOBI POL YNOMIALS
Abstract:
A.K. Chongdar
Department ofMathematics
Bangabasi Evening College
19, R.K. Chakraborty Sarani,
Calcutta-700009
INDIA
and
N.K. Majumdar
Department ofMathematics
Bagnan College, P.O. Bagnan, Dist.- Howrah,
West Bengal, INDIA
In this note we have obtained an integral representation for the product of
two extended Jacobi polynomials. Sorne particular cases of interest are also
pointed out..
l. Introduction
The extended Jacobi polynimials defined by Patil and Thakare [1] are
(1.1) F n (a, í3; x) = (-l)'n (x - arª (b- xr13 (-b'A )n
n. -a
X un [(x - a)n+cx (b - xt+13 ], D = ~­dx
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The aim at presenting the note is to derive an integral representation for the
product of two extended Jacobi polynomials as defined in (1.1). In this
connection we would like to point it out that the integral representation for
the product of two Jacobi polynomials derived by S. K. Chatterjea [2] is
incorrect and the correct version of the same has been obtained as a special
case of our result. In fact, it may be mentioned here that in [2], the relation
(2.9) is wrong and the results (2.13) and (2.14) are incorrect - the correct
versions of which are given in (2.9) and (2.10) of this note.
2. DERIV ATION
In deriving our result we shall make use of the following
formulae [3] :
(2.1)
(2.2)
(2.3)
Sn (a., y, x) = 2F1 (- n, a+ n; y; x)
(1 + P)n n X - b
Fn(a,f3;x)= /..., 2F1 (-n,l+a+f3+n;l+f3;--
n! a-b
(ex)
<l>n (a., y; x) = nf- 2F1 (-n, a+ n; y; x)
(ex)
= -,n sn (a, y; x).
n .
The definitions (2.1), (2.3) are used in [3] and [2]. The relation (2.2) is
proved by the present authors [4].
From (2.3) it follows that
(2.4) ~ ( ) ~ (-1)' (ex)n:, r
-v ay:x=¿.,, x .
n ' ' r=O (n - r)! (Y), r !
Now replacing a by (1 +a+ f3), y by (1 + f3) and x by (:=~)in (2.3), we
ha ve
(2.5)
Therefore, with the help of (2.4), we obtain
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(2.6) F (a,f3;x)F (a',f3';y)=A-m+n (l+P)m(l+P')n
m n (1 + a + P)m (1 + a' + P')n
X l: Í:, (-ly+•(l+a+P)m+.(l+a'+P')n+s
r=O s=O (m-r)! (n-s)! (1 +P), (1 +P').
X _1 _ (X - b)' (I._- b)s.
r! s! a - b a - b
Now using the results [5]:
1t/2
I' (μ+V+ l) = ~ f e(μ-v)9i COS(μ+v) e de (μ+V> -1)
r(μ+l)r(v+l) n -1t/2 '
and
r (μ) r (v) = fl t(μ-1) (1 - t)(v-1) dt, (μ>o, V> O)
r (μ + v) 0
we obtain from (2.6)
(2.7) F m (a, f3; x) Fn (a', f3'; y)
= Am +n __(_ 1+ _P)_m{ _l_+_P'_)n_ _ r (1 + P) r (1 + P')
(1 + a + P)m (1 + a' + P')n r (1 + a + P) r (1 + a' + P')
2 ~+Jl'+m+n 1 7t/2 7t/2
X ---;;-2 - f f f ex+l3+m (1 - t)cx' + 13' + n
o -1t/2 -1t/2
X e(l3- l3') <l>i + (m- n) 0i COS(m+n) e COS(l3 + l3') <l>
(i:n) (-l)k cos <I> k r (2 +a+ P +a'+ W + m + n + k)
X k =o k! ( cose ) r (P + w + k + 1) (m + n - k) !
x L (k) ( t (x - b) )r( (1 - t) (y- b) )s e(r-s) (<I>-S)i d<l> de dt
r+s=k r a - b a - b
(1 + P)m (1 + P')n
=A m+n --------
r (1 + P) r (1 + P')
(1 + a + P)m (1 + a' + P')n r (1 + a + P) r (1 + a' + P')
1 1t/2 1t/2
X I' (2 + Cl + P +a'+ P') _2_~_+~_·+_m_+n-J f f tcx+l3+m
r <P + W + 1) n2 o -1t12 -1t12
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(1 t)a.' + J3' + n (J3-J3') <l>i + (m - n) 8i (m+n) e (J3+J3') ~
X - e COS COS 'V
(m+n) (-l)k
X L
(2 + a + R + a' + R') P P m+n+k
k=O k ! (P + W + l)k (m+n-k) !
{ (a - b) a - b
t .<_x_-_b)_e(<1>-e)i + (1 - t) ( y - b ) e(e-<1>Ji lk
X ------------- COS <1> d<I> de dt
cose
= 1.,<m+n) (l+P)m (l+P')n r (l+P) r (l+W) r (2+a+p+a'+W)
(1 +a+P)m (1 +a'+P')n r (1 +a+p) r (1 +a'+W) r (P+W +l)
' 1 7t/2 7t/2
X 2 (fl+::m+n) f f f ta.+J3+m (1 - tt'+J3'+n
o -7t/2 -7t/2
X e(J3-J3') <l>i+ (m-n) 8i COS(m+n) e COS(J3 + J3') <1> X <1> (2+a+r.t+a' +A'
m+n P P'
t (x-b) y - b
{ e(<1>-e)i +(1-t)( ) e(0--<1>Ji} cos <I>
f3+f3'+1, (a-b) a-b )d<I>dedt
cose
obtained by making use ot (2.4).
Now from (2.7), by using (2.5), we obtain
(2.8) Fm(a,f3;x)Fn(a',f3';y)
X
2 (f3+W+m+n) r (a+ a'+ p + W + m + n + 2)
1t2 r (a+ p + m + 1) r (a' + w +n + 1)
r (P + m + 1) r (P' + n + 1)
r (P + W + m + n + 1)
1 7t/2 7t/2 f f f ta.+J3+m (1 - tt'+J3'+n
o -7t/2 -7t/2
X e(J3-J3') <l>i + (m-n) Si COS(m+n) e COS(J3 + J3') <1>
x F m+n ( a+a' + l,f3+ f3';
{t (x - b) e(<1>-e)i + (1- t) (y- b) e(e-<1>)i} cos <I>
b+ ) d<I> de dt
cose
which is our desired result.
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Sorne special cases :
Case - l. Now putting b = - a = 1 and A, = 1 (in 2.8) and finally
interchanging a , 13 and a', 13', we ha ve
(2.9)
( ") ( , "') 2(cx+cx'+m+n)
p m ex, ,.. (x) p nex ' ,.. (y) = 2
7t
r (a+a'+~+W+m+n+2)
r (a+~+m+l) r (a'+W+n+l)
X
r (a+ m + 1) r (a'+ n + 1)
r (a+ a'+ m + n + 1)
1 7t/2 7t/ 2 f f f tex+l3+m (1 - tt' +13'+n
o -7t/2 -7t/2
X e(ex-ex') <l>i + (m-n) 8i COS(m+n) 8 COS(ex +ex') <l>
, , {t (x-l)e(<1>-e)i+(l-t) (y-l)e(e-<1>)i}cos et>
X p + (ex+ex , 13+13 +l) (1 + - ) d<l>d8dt,
m n cose
which is the correct version of the integral representation of the product
of two jacobi polynomials derived by S. K. Chatterjea in (2.13) of [2].
Case-11 Now putting a= 13=a'=13' =O in (2.9), we have
1 7t/ 2 7t/2
P (x) P (y)= _2'-1_ (m + n + 1) J J J tm
m n 7t2
o -7t/ 2 -7t/ 2
(2.10)
X (1- tte (m-n)8iCOS(m+n) 8
{t (x-1) e(<1>-e)i + (1-t) (y-1) e(0-<1>)i} cos et>
X p (O, l) (1 + ) d<l> d8 dt
m+n cose
which is correct version of (2.14) of [2].
References :
[1] K.R. Patil and N.K. Thakare : Operational formulae and generating functions in
the united form for the classical orthogonal polynomials, the Mathematics
Student, 45(1) (1977), 41-51.
[2] S.K. Chatterjee : Integral representation for the product of two Jacobi
polynomials, Joumal London Math. Soc., 39 (1964), 753-756.
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[3] W. Magnus & F. Oberhettinger : Special Functions of Mathematical Physir,s
(translated by J. Werner), New York, 1954, p.83.
[4] A.K. Chongdar & N.K. Majumdar : Sorne properties of extended Jacobi
polynomials · - Communicated.
[5] E.T. Whittaker & G.N. Watson : A course of modern analysis, 4th Ed.,
Cambridge, 1952, p.263, 253.
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