On the singularities of the continuous Jacobi transform when 𝛼+𝛽=0
Authors: Zayed Ahmed I. 
Published: June 2010

Let

α
,
β
>
−
1

\alpha ,\beta > - 1

and
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper P Subscript lamda Superscript left-parenthesis alpha comma beta right-parenthesis Baseline left-parenthesis x right-parenthesis equals left-parenthesis 1 minus x right-parenthesis Superscript alpha Baseline left-parenthesis 1 plus x right-parenthesis Superscript beta Baseline upper P Subscript lamda Superscript left-parenthesis alpha comma beta right-parenthesis Baseline left-parenthesis x right-parenthesis">

P

λ

(
α
,
β
)

(
x
)
=

(
1
−
x

)
α

(
1
+
x

)
β

P
λ

(
α
,
β
)

(
x
)

\mathcal {P}_\lambda ^{(\alpha ,\beta )}(x) = {(1 - x)^\alpha }{(1 + x)^\beta }P_\lambda ^{(\alpha ,\beta )}(x)

, where

P
λ

(
α
,
β
)

(
x
)

P_\lambda ^{(\alpha ,\beta )}(x)

is the Jacobi function of the first kind,

λ
≥
−
(
α
+
β
+
1
)

/

2

\lambda \geq - (\alpha + \beta + 1)/2

, and

−
1
>
x
≤
1

- 1 > x \leq 1

. Let
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F Superscript left-parenthesis alpha comma beta right-parenthesis Baseline left-parenthesis lamda right-parenthesis equals StartFraction 1 Over 2 Superscript alpha plus beta plus 1 Baseline EndFraction mathematical left-angle f left-parenthesis x right-parenthesis comma script upper P Subscript lamda Superscript left-parenthesis alpha comma beta right-parenthesis Baseline left-parenthesis x right-parenthesis mathematical right-angle equals StartFraction 1 Over 2 Superscript alpha plus beta plus 1 Baseline EndFraction integral Subscript negative 1 Superscript 1 Baseline f left-parenthesis x right-parenthesis script upper P Subscript lamda Superscript left-parenthesis alpha comma beta right-parenthesis Baseline left-parenthesis x right-parenthesis d x">

F

(
α
,
β
)

(
λ
)
=

1

2

α
+
β
+
1

⟨

f
(
x
)
,

P

λ

(
α
,
β
)

(
x
)

⟩

=

1

2

α
+
β
+
1

∫

−
1

1

f
(
x
)

P

λ

(
α
,
β
)

(
x
)
d
x

{F^{(\alpha ,\beta )}}(\lambda ) = \frac {1} {{{2^{\alpha + \beta + 1}}}}\left \langle {f(x),\mathcal {P}_\lambda ^{(\alpha ,\beta )}(x)} \right \rangle = \frac {1} {{{2^{\alpha + \beta + 1}}}}\int _{ - 1}^1 {f(x)\mathcal {P}_\lambda ^{(\alpha ,\beta )}(x)dx}

\]
whenever the integral exists. It is known that for

α
+
β
=
0

\alpha + \beta = 0

, we have (*)
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="f left-parenthesis x right-parenthesis equals limit Underscript n right-arrow normal infinity Endscripts 4 integral Subscript 0 Superscript n Baseline upper F Superscript left-parenthesis alpha comma beta right-parenthesis Baseline left-parenthesis lamda minus one half right-parenthesis upper P Subscript lamda minus 1 slash 2 Superscript left-parenthesis beta comma alpha right-parenthesis Baseline left-parenthesis negative x right-parenthesis lamda times sine pi lamda StartFraction normal upper Gamma squared left-parenthesis lamda plus 1 slash 2 right-parenthesis Over normal upper Gamma left-parenthesis lamda plus alpha plus 1 slash 2 right-parenthesis normal upper Gamma left-parenthesis lamda plus beta plus 1 slash 2 right-parenthesis EndFraction d lamda">

f
(
x
)
=

lim

n
→
∞

4

∫
0
n

F

(
α
,
β
)

(

λ
−

1
2

)

P

λ
−
1

/

2

(
β
,
α
)

(
−
x
)
λ
×
sin
⁡
π
λ

Γ
2

(
λ
+
1

/

2
)

Γ
(
λ
+
α
+
1

/

2
)
Γ
(
λ
+
β
+
1

/

2
)

d
λ

f(x) = \lim \limits _{n \to \infty } 4\int _0^n {{F^{(\alpha ,\beta )}}\left ( {\lambda - \frac {1}{2}} \right )} P_{\lambda - 1/2}^{(\beta ,\alpha )}( - x)\lambda \times \sin \pi \lambda \frac {{{\Gamma ^2}(\lambda + 1/2)}}{{\Gamma (\lambda + \alpha + 1/2)\Gamma (\lambda + \beta + 1/2)}}d\lambda

\]
almost everywhere in

[
−
1
,
1
]

[-1,1]

. In this paper, we devise a technique to continue

f
(
x
)

f(x)

analytically to the complex

z
z

-plane and locate the singularities of

f
(
z
)

f(z)

by relating them to the singularities of
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="g left-parenthesis t right-parenthesis equals integral Subscript 0 Superscript normal infinity Baseline e Superscript minus lamda t Baseline upper F Superscript left-parenthesis alpha comma beta right-parenthesis Baseline left-parenthesis lamda right-parenthesis StartFraction d lamda Over normal upper Gamma left-parenthesis lamda plus alpha plus 1 right-parenthesis EndFraction period">

g
(
t
)
=

∫
0
∞

e

−
λ
t

F

(
α
,
β
)

(
λ
)

d
λ

Γ
(
λ
+
α
+
1
)

.

g(t) = \int _0^\infty {{e^{ - \lambda t}}{F^{(\alpha ,\beta )}}(\lambda )} \frac {{d\lambda }}{{\Gamma (\lambda + \alpha + 1)}}.

\]
However, this will be done in the more general case where the limit in (*) exists in the sense of Schwartz distributions and defines a generalized function

f
(
x
)

f(x)

. In this case, we pass from

f
(
x
)

f(x)

to its analytic representation
\[
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove f With caret left-parenthesis z right-parenthesis equals StartFraction 1 Over 2 pi i EndFraction mathematical left-angle f left-parenthesis x right-parenthesis comma StartFraction 1 Over x minus z EndFraction mathematical right-angle comma z not-an-element-of s u p p f comma">

f
^

(
z
)
=

1

2
π
i

⟨

f
(
x
)
,

1

x
−
z

⟩

,

z
∉
supp
⁡
f
,

\hat f(z) = \frac {1} {{2\pi i}}\left \langle {f(x),\frac {1} {{x - z}}} \right \rangle ,\quad z \notin \operatorname {supp} f,

\]
and then relate the singularities of

f
^

(
z
)

\hat f(z)

to those of

g
(
t
)

g(t)

.