# Gegenbauer polynomials revisited

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GEGENBAUER POLYNOMIALS REVISITED
A. F. HORADAM
University

of New England, Armidale,

Australia

(Submitted June 1983)

1. INTRODUCTION
The Gegenbauer (or ultraspherical)
polynomials

Cn(x) (A > -%, \x\ < 1) are

defined by
c\(x)

= 1, c\(x)

= 2Xx

(1.1)

with the recurrence relation
nC„{x)

= 2x(X + n - 1 ) < ^ - I O 0 - (2X + n - 2)CnA_2(^)

(w > 2) .

(1.2)

Gegenbauer polynomials are related to Tn(x),
the Chebyshev polynomials
of
the first kind, and to Un(x), the Chebyshev polynomials of the second kind, by the relations
Tn (x) = | liml—JJ—I

(n>l),

(1.3)

and

tffe)= Cite).

(1.4)

Properties of the rising and descending diagonals of the Pascal-type arrays of {Tn(x)}
and {Un(x)} were investigated in , , and , while in  the rising diagonals of the similar array for C^(x) were examined. Here, we consider the descending diagonals in the Pascal-type array for {Cn(x)}9

with a backward glance at some of the material in .
As it turns out, the descending diagonal polynomials have less complicated computational aspects than the polynomials generated by the rising diagonals. Brief mention will also be made of the generalized Humbert polynomial, of which the Gegenbauer polynomials and, consequently, the Chebyshev polynomials, are special cases.

2.

DESCENDING DIAGONALS FOR THE GEGENBAUER POLYNOMIAL ARRAY

Table 1 sets out the first few Gegenbauer polynomials (with y = 2x):

294

[Nov.

GEGENBAUER POLYNOMIALS REVISITED
TABLE 1.

Descending Diagonals for Gegenbauer Polynomials

(2.1)

dhx)

CUx)
ds(x)
C)(x)
wherein we have written
(X + n - 1).

(A)„ = X(X + 1)(X + 2)

(2.2)

Polynomials (2.1) may be obtained either from the generating recurrence (1.2) together with the initial values (1.1), or directly from the known explicit summation representation

m= 0

ml (n - 2m)!

—, X an integer and n ^ 2,

wherej, as usual, [n/2] symbolizes the integer part of

(2.3)

n/2.

The generating function for the Gegenbauer polynomials is

E C^(x)tn

= (1 - 2xt + t2)~X

(\t\

< 1).

(2.4)

n=0

Designate the descending diagonals in Table 1, indicated by lines, by the symbols dj(x)
(j = 0, 1, 2, ...) .
Then we have

d0(x)

= 1, d^{x)

= X(2a; - 1), d 2(x) =

(A)3 ) =

(X),(2ar - 1) :

x

;

2!

(A^Ote - l)'

> - - ^ n

X +

1

r

)(2x-D",

(2.6)

a result which we now proceed to prove.
Proof of (2.6): Suppose we represent the pairs of values of m and n which give rise to d^(x) by the couplet (m9 n).
Then, at successive "levels" of the descending diagonal dn(x) we have the couplets

in Table 1,

(0, n ) , (1, n + 1), (2, n + 2 ) , ..., (n - 1, 2n - 1), (n, 2n), so that corresponding values of n - 2m are n, n - 1, n - 2, ..., 19 0, while n - m always has the value n.
[It is important to note that the maximum value
for m in the couplets must be n.]
Consequently, from Table 1 and (2.3), with y = 2x for convenience, we have

(A)y*-1

a)y»
i A /

U {X)

v

n

_

"

"

0!n!

n

__

wy-2

,

l!(n - 1)!

^

wy
_

2!(n - 2)!

,

(_\

^

=^-""=^

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