Differential-difference properties of generalized Jacobi polynomial

Wimp, J.

English

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D I F F E R F i N T L C Y L - D I F N m  PROPERTIES OF 
GElXEFULLIZED JACOB1 I? 
ImIM FE?J?CIRT 
15 January 1969 
Contract No. NAS9-7641 
MRI Project No. 3162-P 
For 
NASA Manned Spacecraft Center 
General Research Procurement Branch 
Houston, Texas 77058 
A t t n  : J .W. Carlson/BG731(48) 
425 VOLKER BOULEVARD, KANSAS CITY, ISSOURI 641 1 0 AREA 816 561-0202 
https://ntrs.nasa.gov/search.jsp?R=19690007624 2020-03-12T07:18:52+00:00Z
D I m W A L - D I F N a  mOPERTIES O F  
CSlE€KUzED JACOB1 POLYNc%IIIAIS 
bY 
J e t  Wimp 
ImIM RFlPORT 
15 January 1969 
Contract No. NAS9-7641 
MRI Project No. 3l.62-P 
For 
NASA Manned Spacecraft Center 
General Research Procurement Branch 
Houston, Texas 77058 
Attn : J . W . C a r  lson/BG73 1 (48 ) 
425 VQLKER BOULEVARD, KANSAS CITY, ISSQURI 64110 AREA 816 561-0202 
PliEFACE 
This report ,  wri t ten by Je t  Wimp, Analysis and Applied Mathe- 
matics Section, Midwest Research Ins t i tu te ,  covers work performed from 
1 Ju ly  1968 t o  1 7  January 1969 on Contract No. NUS-7645. 
Approved for : 
MIIWEST RESEARCH INSTITUTE 
Harold Stout, Director 
Engineering Sciences Division 
17  January 1969 
l w  
ii 
TABLE OF CONTENTS 
Page No. 
I. Introduction . . . . . . . . . . . . . . . . . . . . . . .  1 
11. Results . . . . . . . . . . . . . . . . . . . . . . . . .  1 
111. Concluding Remarks . . . . . . . . . . . . . . . . . . . .  6 
R e f e r e n c e s . .  . . . . . . . . . . . . . . . . . . . . . . . . .  7 
ii i 
I. IN!LBODUCTION 
In  t h i s  paper we derive a differential-difference equation satis- 
f ied  by the hypergeometric polynomials 
Throughout, we employ the shorthand notation 
P 
j =1 
(ap+n) = (aj+n) , etc., 
see [l]. In  general, where any variable i s  subscripted by a p or q , it 
i s  t o  be understood t h a t  the shorthand notation has been invoked. 
11. RESULTS 
Theorem 
ii) none of the quantit ies bj, A, A+l-bj be negative 
integers or zero, j = 1,2,. . . ,q ; 
- iii) no bj = any ah , h = 1,2 ,..., p ; j - 1,2 ,..., q .  
Then the polynomials Pn(x) s a t i s fy  the differential-difference equation 
where 
l 
and no such equation of lower order CT' < y exists. The A v ' s  and By's 
are unique and 
% =  x , v > o ;  
c 
( - ~ ) ~ ( n - s ) ( b  +n-s-1) 
s=o E ( V + S - ~ ~ - A ) ~ ~ - ~  
B v =  I 
- 6 n , v = O  . I 
Proof: 
r 
j V ' O J  
v - 1  (l-v),(a +n-s-l) 
x z  s=o ( v + s + ~ - ~ - 2 n  )a-v 
i n  (3) we f ind 
k+l 
By equating coefficients of x 
(5 1 
where 
2 
The above can be considered an ident i ty  between polynomials i n  
the (generally complex-valued) variable k . I f  p+l  > q , ( 7 )  requires 
t ha t  two polynomials of degree p+&2 be identical ,  and t h i s  condition 
furnishes p+&3 equations i n  20+2 unknowns so we must have 0 S p+l . 
If p+l = q , we similarly f ind 0 5 p+l  , while i f  p+l C q , we f ind tha t  
o 5 q . Thus 
Now i f  we assume equality above, the  4 and B, ( i f  they ex i s t )  are 
unique. Suppose there  is  another such recurrence re la t ion  with coefficients 
and B.J” . Subtracting these two, we have 
but t h i s  i s  impossible, under the hypothesis (ii) and (iii), since Wimp has 
shown t h a t  i n  t h i s  case any l inear  difference equation sa t i s f i ed  by 
must be of order 0+l at least ,  see [l]. 
Pn(x) 
Now, i f  q = p+1 , ( 7 )  holds if and only i f  
(Note tha t  a sui table  l inear  combination of (11) and (E)  gives ( 7 ) ,  i.e., 
multiply (11) by (k-n)(n+h+k-o)(ap+k) and (l2) by (k+l)(bq+k) and add.,) 
To establ ish (11) for  
tween t w o  polynomials i n  k , each of degree q+2 and each having two 
ident ica l  factors.  It only remains t o  shm (11) holds for q+l d is t inc t  
values of k . 
p+l = q , we observe tha t  it asser ts  an ident i ty  be- 
Assume a l l  the quantit ies -1, -bj, j = 1,2, ..., q , are 
3 
d i s t inc t  and l e t  k have these values i n  (7) .  The r e su l t  is  (11) evaluated 
a t  these values. 
Xkewise t o  show (E) f o r  p+l = q , we need only prove it holds 
f o r  the 
is  true, since ( 7 )  and (E)  for  these values me the same. 
p+2 values (assumed d i s t inc t )  n, o-n-A, -aj, j = 1,2 ,..., p . This 
(The requirement tha t  the values of k chosen above be d i s t inc t  
may be relaxed by continuity.) 
Now replace x by "/aj, j = p'+1, p1+2, ...,q- 1 i n  (3), where 
p'  C q-1 . This shows tha t  
where PA(.) i s  Pn(x) with p replaced by p '  and 
The same l i m i t  process applied t o  (E)  yields the following equa- 
t i on  for the determination of D$ : 
(T 
(k-n)(n+A+k-o)(a ,+k) = D$c!,(k)B,(k) 
v=o P 
The equation for C, i n  t h i s  case is  (11) as it stands. But 
(11) and. (15) together are (11) and (E), respectively, wri t ten for 
p + l <  q . 
Similarly, replacing x by xbJ ,  j = q't-1, q'+2 ,..., p+l, q' S p 
and l e t t i ng  b.+= i n  (3) gives 
J 
4 
where 
and P:(x) is Fn(x) with q replaced by q'  . This l i m i t  process ap- 
plied t o  (11) gives 
and (12) i s  used unchanged for  
(11) and (E?) for 
Dv . These two equations, though, are just 
p+l  > q . 
Thus (11) and (E) are established for a l l  p,q and w e  have suc- 
ceeded i n  "uncoupling" Eq. ( 7 )  to give Eqs. (11) and (E) which involve 
and Dv alone, respectively. 
C, 
N e x t ,  we solve these two equations. 
In Eq. (ll), le t  k+l-n = -s , s = 0,1,2 ,..., 5 . The resu l t  can 
be wri t ten 
s = 0,1,2 ,...) CJ . (19 1 
But if 1-2n-h # 0,-1,-2,. . ., the  above equation can be solved f o r  
applying a Lemma of Wimp [l] . 
of uni t  argument, one arrives a t  (5). To f ind  the &Is, l e t  k-n = -s i n  
(E) and proceed i n  a similm fashion. 
C, by 
2F1'~ After some algebra and evaluation of 
5 
111. CONCLUDING REMARKS 
If p+ l  = q and x = 1 i n  (3), we get  a recursion re la t ion  f o r  
Pn(S) of order max(p+l,q). Note tha t  t h i s  i s  of order one less than t h a t  
obtained by putting i n  the homogeneous l inear  difference equation 
sa t i s f i ed  by Pn(x) given i n  pJ. If p = 1 , the  resul t ing recursion re-  
la t ion  f or 
x = 1 
is t h a t  given by BaileyL31, which i n  turn  is Watson’s resu l t  [2] s l i gh t ly  
rewritten. For p+l  = q and x general, (3) of course provides a gener- 
a l izat ion of the c lass ica l  differential-difference formula f o r  the Jacobi 
polynomials, see [4, p. 170 (15)]. 
A differential-difference re la t ion  f o r  the polynomials 
can be eas i ly  obtained from (3) by replacing x by x/h and l e t t i ng  
h+W. 
We point out t ha t  the conditions of the theorem can be relaxed 
considerably. If h is  a posit ive integer m , we can w r i t e  
which is well-defined for  a l l  
analysis. ~ 
n , s o  condition (i) i s  not essent ia l  t o  the  
Also, i f  any of the quantit ies (ii) are negative integers or zero, 
limits may be taken after the equation has been multiplied by a suitable 
factor,  see [l]. The quantity n can even be nonintegral when q > p f l  
or when q = p+l  and larg(1-x) I < IT , by the permanence principle f o r  
functional equations. (It may be necessary i n  t h i s  case t o  multiply the  
equation by a factor  (r-n-h) t o  make the  coefficients well-defined.) 
6 
11 1. Wimp, Je t ,  Recursion Formulae f o r  Hypergeometric Functions, 'I Math. 
ComE., 2- (1968) 363-373. 
2. Watson, G.N., "A Reduction Formula," Proc. Glasgm Math. Assoc., R 
(1954) 57-61. 
3. Bailey, W.N., "Contiguous I-lypergeometric Functions of the Ty-pe 
gF2(1)," Proc. Glasgm Math. Assoc., N 2 (1954) 62-65. 
4. Erde'lyi, A., W. Magnus, F. Oberhettinger, and F.G. Tricomi, Higher 
Transcendental Functions, v. 2, Ch. X, McGraw-Hill, New York (1953). 
7 


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