Asymptotic formulas for ultraspherical polynomials 𝑃_{𝑛}^{𝜆}(𝑥) and their zeros for large values of 𝜆
Authors: Elbert Árpád , Laforgia Andrea 
Published: June 2010

For

λ
>
−
1

/

2

\lambda > - 1/2

we denote by

P
n

(
λ
)

(
x
)

P_n^{(\lambda )}(x)

the ultraspherical polynomial of degree

n
n

and by

x

n
,
k

(
λ
)

x_{n,k}^{(\lambda )}

and

h

n
,
k

(
k
=
1
,
2
,
…
,
n
)

{h_{n,k}}(k = 1,2, \ldots ,n)

the

k
k

th zeros of

P
n

(
λ
)

(
x
)

P_n^{(\lambda )}(x)

and of the Hermite polynomial

H
n

(
x
)

{H_n}(x)

, respectively. In this paper we establish the following formulas
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="lamda Superscript negative n slash 2 Baseline upper P Subscript n Superscript left-parenthesis lamda right-parenthesis Baseline left-parenthesis StartFraction x Over StartRoot lamda EndRoot EndFraction right-parenthesis equals sigma-summation Underscript j equals 0 Overscript n minus 1 Endscripts lamda Superscript negative j Baseline upper Q Subscript n j Baseline left-parenthesis x right-parenthesis for lamda not-equals 0">

λ

−
n

/

2

P
n

(
λ
)

(

x

λ

)

=

∑

j
=
0

n
−
1

λ

−
j

Q

n
j

(
x
)

for

λ
≠
0

{\lambda ^{ - n/2}}P_n^{(\lambda )}\left ( {\frac {x}{{\sqrt \lambda }}} \right ) = \sum \limits _{j = 0}^{n - 1} {{\lambda ^{ - j}}{Q_{nj}}(x)\,{\text {for}}\,\lambda \ne 0}

\]
and
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x Subscript n comma k Superscript left-parenthesis lamda right-parenthesis Baseline equals h Subscript n comma k Baseline lamda Superscript negative 1 slash 2 Baseline minus StartFraction h Subscript n comma k Baseline Over 8 EndFraction left-parenthesis 2 n minus 1 plus 2 h Subscript n comma k Superscript 2 Baseline right-parenthesis lamda Superscript negative 3 slash 2 Baseline plus h Subscript n comma k Baseline left-parenthesis StartFraction 12 n squared minus 12 n plus 1 Over 128 EndFraction plus StartFraction 5 n minus 2 Over 24 EndFraction h Subscript n comma k Superscript 2 Baseline plus five ninety-sixths h Subscript n comma k Superscript 4 Baseline right-parenthesis lamda Superscript negative 5 slash 2 Baseline plus upper O left-parenthesis lamda Superscript negative 7 slash 2 Baseline right-parenthesis comma lamda right-arrow normal infinity">

x

n
,
k

(
λ
)

=

h

n
,
k

λ

−
1

/

2

−

h

n
,
k

8

(
2
n
−
1
+
2

h

n
,
k

2

)

λ

−
3

/

2

+

h

n
,
k

(

12

n
2

−
12
n
+
1

128

+

5
n
−
2

24

h

n
,
k

2

+

5

96

h

n
,
k

4

)

λ

−
5

/

2

+
O
(

λ

−
7

/

2

)
,
λ
→
∞

x_{n,k}^{(\lambda )} = {h_{n,k}}{\lambda ^{ - 1/2}} - \frac {{{h_{n,k}}}}{8}(2n - 1 + 2h_{n,k}^2){\lambda ^{ - 3/2}} + {h_{n,k}}\left ( {\frac {{12{n^2} - 12n + 1}}{{128}} + \frac {{5n - 2}}{{24}}h_{n,k}^2 + \frac {5}{{96}}h_{n,k}^4} \right ){\lambda ^{ - 5/2}} + O({\lambda ^{ - 7/2}}),\lambda \to \infty

\]
where

Q

n
0

(
x
)
=

H
n

(
x
)

/

n
!

{Q_{n0}}(x) = {H_n}(x)/n!

and

Q

n
j

(
x
)
(
j
=
1
,
2
,
…
,
n
−
1
)

{Q_{nj}}(x)(j = 1,2, \ldots ,n - 1)

are polynomials specified in Theorem 1. Finally we show that the positive (negative) zeros of

P
n

(
λ
)

(
x
)

P_n^{(\lambda )}(x)

are convex (concave) functions of

λ
\lambda

, provided

λ
\lambda

is sufficiently large.