A new property of the Wallis power function
Authors: Yang Zhen-Hang , Tian Jing-Feng 
Published: November 2023

The Wallis power function is defined on

(
−
min

{
p
,
q
}

,
∞
)

\left ( -\min \left \{ p,q\right \},\infty \right )

by
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper W Subscript p comma q Baseline left-parenthesis x right-parenthesis equals left-parenthesis StartFraction normal upper Gamma left-parenthesis x plus p right-parenthesis Over normal upper Gamma left-parenthesis x plus q right-parenthesis EndFraction right-parenthesis Superscript 1 slash left-parenthesis p minus q right-parenthesis Baseline if p not-equals q and upper W Subscript p comma p Baseline left-parenthesis x right-parenthesis equals e Superscript psi left-parenthesis x plus p right-parenthesis Baseline period">

W

p
,
q

(
x
)

=

(

Γ

(
x
+
p
)

Γ

(
x
+
q
)

)

1

/

(
p
−
q
)

 if 
p
≠
q
 and 

W

p
,
p

(
x
)

=

e

ψ

(
x
+
p
)

.

\begin{equation*} W_{p,q}\left ( x\right ) =\left ( \dfrac {\Gamma \left ( x+p\right ) }{\Gamma \left ( x+q\right ) }\right ) ^{1/\left ( p-q\right ) }\text { if }p\neq q\text { and }W_{p,p}\left ( x\right ) =e^{\psi \left ( x+p\right ) }. \end{equation*}

Let

p

i

,

q

i

∈

R

p_{i},q_{i}\in \mathbb {R}

with

0
>

δ

i

=

p

i

−

q

i

≤
1

0>\delta _{i}=p_{i}-q_{i}\leq 1

,

θ

i

=

(
1
−

δ

i

)

/

2

\theta _{i}=\left ( 1-\delta _{i}\right ) /2

for

i
=
1
,
2

i=1,2

and

p

1

+

q

1

=

p

2

+

q

2

=
2
σ
+
1

p_{1}+q_{1}=p_{2}+q_{2}=2\sigma +1

. We prove that, if

q

1

>

q

2

q_{1}>q_{2}

then for any integer

m
∈

N

m\in \mathbb {N}

, the function
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="x right-arrow from bar left-parenthesis negative 1 right-parenthesis Superscript m Baseline left-bracket ln StartFraction upper W Subscript p 1 comma q 1 Baseline left-parenthesis x right-parenthesis Over upper W Subscript p 2 comma q 2 Baseline left-parenthesis x right-parenthesis EndFraction minus sigma-summation Underscript k equals 1 Overscript m Endscripts StartFraction a Subscript 2 k Baseline left-parenthesis theta 1 comma theta 2 right-parenthesis Over left-parenthesis 2 k plus 1 right-parenthesis left-parenthesis 2 k right-parenthesis left-parenthesis x plus sigma right-parenthesis Superscript 2 k Baseline EndFraction right-bracket">

x
↦

(
−
1
)

m

[
ln
⁡

W

p

1

,

q

1

(
x
)

W

p

2

,

q

2

(
x
)

−

∑

k
=
1

m

a

2
k

(

θ

1

,

θ

2

)

(
2
k
+
1
)

(
2
k
)

(
x
+
σ
)

2
k

]

\begin{equation*} x\mapsto \left ( -1\right ) ^{m}\left [ \ln \frac {W_{p_{1},q_{1}}\left ( x\right ) }{W_{p_{2},q_{2}}\left ( x\right ) }-\sum _{k=1}^{m}\frac {a_{2k}\left ( \theta _{1},\theta _{2}\right ) }{\left ( 2k+1\right ) \left ( 2k\right ) \left ( x+\sigma \right ) ^{2k}}\right ] \end{equation*}

is completely monotonic on

(
−
σ
,
∞
)

\left ( -\sigma ,\infty \right )

, where
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="a Subscript 2 k Baseline left-parenthesis theta 1 comma theta 2 right-parenthesis equals StartFraction upper B Subscript 2 k plus 1 Baseline left-parenthesis theta 2 right-parenthesis Over theta 2 minus 1 slash 2 EndFraction minus StartFraction upper B Subscript 2 k plus 1 Baseline left-parenthesis theta 1 right-parenthesis Over theta 1 minus 1 slash 2 EndFraction period">

a

2
k

(

θ

1

,

θ

2

)

=

B

2
k
+
1

(

θ

2

)

θ

2

−
1

/

2

−

B

2
k
+
1

(

θ

1

)

θ

1

−
1

/

2

.

\begin{equation*} a_{2k}\left ( \theta _{1},\theta _{2}\right ) =\frac {B_{2k+1}\left ( \theta _{2}\right ) }{\theta _{2}-1/2}-\frac {B_{2k+1}\left ( \theta _{1}\right ) }{\theta _{1}-1/2}. \end{equation*}

This extends and generalizes some known results.