Revista Brasileira de Física, Vol. 10, NP 1, 1980
The Green's Function for Jacobi Special Functions
J. BELLANDI FILHO and E. CAPELAS DE OLIVEIRA*
Instituto de Física, Universidade Estadual de Campinas, 13160 Campinas, SP.
Recebido em 4 de Outubro de 1979
The Green's f u n c t i o n f o r the Jacobi d i f f e r e n t i a l equation
As by -product
i s c a l c u l a t e d hy means o f t h e S t u r m - L i o u v i l l e method.
we o b t a i n Legendre, Gegenbauer and Tchebichef Green's f u n c t i o n s .
Calcula- se a função de Green para a equação d i f e r e n c i a l de
Jacobi p e l o método de S t u r m - L i o u v i l l e . Como casos p a r t i c u l a r e s obtém-se também as funções de Green para as equações d i f e r e n c i a i s
de
Le-
gendre, Gegenbauer e Tchebichef.
1. INTRODUCTION
The present
paper contains
a sistematic
calculation o f
t h e S t u r m - L i o u v i l l e expansion o f the Green's f u n c t i o n f o r the d i f f e r e n t i a l equation f o r a number o f special f u n c t i o n s ;
a common f e a t u r e
f o r a11 o f them i s t h a t the corresponding d i f f e r e n t i a l
be d e r i v e
from the Jacobi
o r hypergeometric
equation can
d i f f e r e n t i a l equatiori.
These i n c l u d e Legendre, Gegenbauer and Tchebichef f u n c t i o n s .
I n the f i r s t
s e c t i o n we c a l c u l a t e
the Green's f u n c t i o n s
f o r the Jacobi d i f f e r e n t i a l equation by the Sturm-Liouvi I l e methodand
f o r d e f i n e general s p h e r i c a l
harmonics i n o r d e r t o
simpl i f y t h e ex-
pression f o r the Green's f u n c t i o n .
I n the second
s e c t i o n we d e r i v e f rom the Jacobi Green's
f u n c t i o n p a r t i c u l a r cases, as Legendre, Gegenbauer
and
Tchebichef
Green's f u n c t i o n , and i n the t h i r d s e c t i o n we present our discussions.
*
With a FAPESP
- São Paulo - B r a s i l - Fel lowship
2. JACOBI GREEN'S FUNCTION
The Jacobi d i f f e r e n t i a 1 operatorl can be w r i t t e n
in
the
f o l l o w i n g way
where the convenient choice o f combinations @+aand
6-a f o r
t e r s o f t h e d i f f e r e n t i a l o p e r a t o r i s o n l y introduced
parame-
t o simplify the
notation.
I n o r d e r t o c a l c u l a t e the Green's f u n c t i o n we f i r s t take
~ , derive the
o u t the weight f u n c t i o n s ( l - ~ ) ~ ( l + x )and
Green's
func-
t i o n f o r the f o l l o w i n g d i f f e r e n t i a l operator
where n = v-a-6.
The Green's f u n c t i o n f o r t h i s d i f f e r e n t i a l o p e r a t o r s a t i s f i e s the f o l l o w i n g inhomogeneous d i f f e r e n t i a l equation
which i s bounded on O
2
x <
m;
here a
7
- 1 and 6 > - 1 i n o r d e r t o ma-
ke the weight f u n c t i o n non-negative and i n t e g r a b l e , b u t the formal relations are v a l i d without t h i s r e s t r i c t i o n .
The Sturm-Liouvi ll e method2 c o n s i s t s t o w r i t e the Green's
f u n c t i o n as the product o f two l i n e a r l y independent s o l u t i o n s o f the
corresponding homogeneous d i f f e r e n t i a l equation, L X J , = O .
These so-
l u t i o n s f o r the Jacobi 'operator a r e P ( 2 a s 2 6 ) ( x ), r e g u l a r a t the o r i -
n
gen and Qn ( 2 a s 2 6 ) ( z ) , r e g u l a r a t the i n f i n i t y .
The s o l u t i o n o f Eq. 3 i s
(2a,2f3)
(x<) Qn
where
(x<)
x< and x> a r e the l e s s e r and g r e a t e r o f x and x ' r e s p e c t i v e l y .
The Green's f u n c t i o n f o r t h e LT, o p e r a t o r i s e a s i l y o b t a i d
ned, i n c o r p o r a t i n g t h e weight f u n c t i o n i n Eq. 4, and w r i t i n g m v - a - 0
We simpl i f y t h i s expression, i n t r o d u c i ng general
spheri
-
c a l harmonics, as used by V i l e n k i n 3 t o analyse representations o f t h e
group Q U ( ~ )o f u n i -modular quasi - uni t a r y rnatrices of the second ordei-.
These f u n c t i o n s a r e s o l u t i o n s o f t h e homogeneous d i f f e r e n t i a l equation
f o r the L o p e r a t o r and a r e d e f i n e d i n terms o f Jacobi f u n c t i o n by
x
where we have used the M i l l e r ' s n o r m a l i z a t i o n 4 . The Green's
f o r t h e L o p e r a t o r i n terms of general s p h e r i c a l harrnonics
x
3. PARTICULAR CASES
a) Legendre Green's function
The Green's f u n c t i o n f o r t h e associa ted Legendre
di f f e-
r e n t i a l equation can be d e r i v e d from the Jacobi Green's f u n c t i o n í f w e
p u t i n Eq. 8 a=b=m/2. Relations between Jacobi and Legendre functions
can be e a s i l y obtained, expanding Jacobi f u n c t i o n s i n terms of hypergeometric functionsl.
These r e l a t i o n s a r e
and the r e l a t i o n s w i t h general s p h e r i c a l harmonics a r e
The associated Legendre Green's f u n c t i o n i s
t h a t i s the same as c a l c u l a t e d by means o f the i s o t r o p i c harmonic osc i l l a t o r Green's f u n c t i o n 5 . T h i s r e s u l t can be a l s o obtained,
apply d i r e c t l y the Sturm-Liouvi 1 l e method t o the Legendre
d i f f e r e n t i a l equation,
because
c &V
and
if
we
associated
a r e two 1 i n e a r l y indepen-
dent s o l u t i o n s o f the homogeneous d i f f e r e n t i a l equation,
regular
at
the o r i g i n and a t the i n f i n i t y r e s p e c t i v e l y .
bl Gegenbauer Green's function
Gegenbauer f u n c t i o n s l a r e
constant mul t i p l ies
functions w i t h a+B = X-1/2 and B-u=O, w i t h X
7
of Jacobi
-1/2 i n o r d e r t o have
a r e a l and i n t e g r a b l e w e i g h t f u n c t i o n s .
Relations
s p h e r i c a l harmonics and Gegenbauer f u n c t i o n s can be
u s i n g r e l a t i o n s between
Gegenbauer and Legendre
between
general
easi l y o b t a i n e d
functionsl
and
are
B
where c B ( x ) and Da(x) a r e t h e f i r s t and second Gegenbauer f u n c t i o n r
a
r e s p e c t i v e l y The Gegenbauer Green's f u n c t i o n i s
.
C) Tchebichef Green's function
There
a r e two s p e c i a l
mul t í p l e s o f Jacobi f u n c t i o n s ;
~ c h e b i c h e f lf u n c t i o n s which a r e
one w i t h 6-a=O, a+B=-1/2
k i n d T ( x ) f u n c t i o n and o t h e r w i t h B-a=O,
n
k i n d U (x) function.
n
a+B= +1/2
for
f o r the f i r s t
the
For t h e f i r s t k i n d f u n c t i o n we have one r e l a t i o n
yn(x)
and e " )
second
between
(x)
and t h e r e l a t i o n w i t h t h e f i r s t g e n e r a l s p h e r i c a l harmonic i s
The second s o l u t i o n f o r t h e f i r s t k. i n d f u n c t i o n
def i n e d i n terms o f t h e Q ( - 1 / 2 , - 1 / 2 ) ( x ) ,
v+1/2
can
be
expand i n g i n terms o f hyper-
geometric functionsl.
I f we c a l 1 Vv+1/2(x)
(-112,-1/21
is
lation w i t h
where Vv+,,2(x)
the second s o l u t i o n there-
i n terms o f hypergeometric f u n c t i o n s i s g i v e n by
The general ' s p h e r i c a l harmonic i n terms Vue1/2(d i s
The Green's f u n c t i o n f o r t h e f i r s t k i n d Tchebichef
func-
tion is
For t h e second k i n d U ( x ) , we can d e r i v e the r e l a t i o n b e t -
n
ween general s p h e r i c a l harmonics and Tchebichef f u n c t i o n s
i n comple-
t e l y analogy w i t h t h e f i r s t k i n d case
where
X ~ - ~ , ~ ( X i)s the second s o l u t i o n and i n terms
t r i c functions we have
of
hypergeome-
and t h e Green's f u n c t i o n f o r t h e second k i n d f u n c t i o n i s
G ( x , x ~ ) = (s2-1)1/4
( ~ ~ ~I/4
- u1 v )- 1 ~ 2 ( x < ~ x v - 1 ~ 2 ( x , ) .
(26)
4. DISCUSSIONS
We have p r e s e n t e d a g l o b a l d e r i v a r i o n
t i o n , bounded on t h e domain O f x <
m,
for
o f t h e Green's f u n c -
a number o f s p e c i a l f u n c -
t i o n s a r e a l s o bounded i n t h e domain -1 f x $ 1, w i t h t h e
t h a t v rnust be an e n t i r e number,
restriction
i n o r d e r t o have polynornial s o l u t i o n s
f o r t h e c o r r e s p o n d i ng homogeneous d i f f e r e n t i a 1 e q u a t i o n .
There a r e many problems i n p h y s i c s , where these d i f f e r e n t i a 1 e q u a t i o n s appear. One example, i s t h e quanturn mechanics Coulombproblem, i n t h e momentum space, where t h e c o r r e s p o n d i n g r a d i a l
equation i s
a Gegnebauer d i f f e r e n t i a 1 equat ion 6 , wh i ch g i v e s t h e e x p r e s s i o n
of
the
r a d i a l Coulomb Green's f u n c t i o n i n terms o f t h e p r o d u c t o f two independ e n t Gegenbauer f u n c t i o n s
.
The quantum mechanics symmetric top 7 ,
where we have a Jacobi d i f f e r e n t i a l e q u a t i o n .
i s another exampl e ,
I f we w r i t e
t i a 1 e q u a t i o n i n E u l e r ' s angles, 0, $ and J, and s e p a r a t e
the d i f f e r e n t i a l equation i n the
the differenthe variables,
0 v a r i a b l e i s a Jacobi d i f f e r e n t i a l e-
q u a t i o n , and t h e Green's f u n c t i o n i s g i v i n g i n terms o f t h e g e n e r a l spher i c a 1 harrnonics b
E q . (8)
,
bounded i n t h e domai n -1 5 x 5 1
.
This equation i s a l s o the d i f f e r e n t i a l equation f o r irreduc i b l e r e p r e s e n t a t ons o f t h e r o t a t i o n group8, and general s p h e r i c a l h a r monics a r e t h e so u t i o n s o f t h i s d i f f e r e n t i a l e q u a t i o n
-I . ? x $ 1.
in
t h e domain
I t i s i n t e r e s t i n g t o n o t e t h a t t h e s e Green's f u n c t i o n s h e r e
c a l c u l a t e d can be also d e r i v e d by means o f t h e harmonic o s c i ll a t o r Green's
f u n c t i o n i n t h e momentum space, and as a b y - p r o d u c t we can d e r i v e d i n t e g r a l r e p r e s e n t a t i o n s f o r t h e Green's f u n c t i o n s and some a d d i t i o n s t h e o renis f o r t h e g e n e r a l s p h e r i c a l harmonics. These c a l c u l a t i o n s
are object
o f another p u b l i c a t i o n .
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