Gegenbauer polynomials revisited

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 [2], [3], and [5], while in [4] 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 [4].
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):

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[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