Definite integral of a hypergeometric function Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Bounding a polynomial from belowSimplifying real part of hypergeometric function with complex parametersQuestion about Meijer-G definition and identityIntegral with incomplete gamma functionProof of Integral (Gradshteyn & Ryzhik 3.462.1)Challenging definite integral of hypergeometric functionsDefinite integral of exponential, power and Bessel function : a hypergeometric function?Double Gaussian definite integral with one variable limitQuestion over limit of integrand for definite integral.Sum involving hypergeometric 2F2 functionBounding a polynomial from below
How many time did Arya actually used needle?
Why is it faster to reheat something than it is to cook it?
Converted a Scalar function to a TVF function for parallel execution-Still running in Serial mode
What is "gratricide"?
What is the appropriate index architecture when forced to implement IsDeleted (soft deletes)?
Do wooden building fires get hotter than 600°C?
How would a mousetrap for use in space work?
What do you call the main part of a joke?
How to improve on this Stylesheet Manipulation for Message Styling
How come Sam didn't become Lord of Horn Hill?
What does Turing mean by this statement?
How do living politicians protect their readily obtainable signatures from misuse?
Is there any word for a place full of confusion?
How does light 'choose' between wave and particle behaviour?
Why weren't discrete x86 CPUs ever used in game hardware?
Would it be easier to apply for a UK visa if there is a host family to sponsor for you in going there?
Putting class ranking in CV, but against dept guidelines
How to get all distinct words within a set of lines?
Why do early math courses focus on the cross sections of a cone and not on other 3D objects?
Project Euler #1 in C++
What initially awakened the Balrog?
How were pictures turned from film to a big picture in a picture frame before digital scanning?
Exposing GRASS GIS add-on in QGIS Processing framework?
Triggering an ultrasonic sensor
Definite integral of a hypergeometric function
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Bounding a polynomial from belowSimplifying real part of hypergeometric function with complex parametersQuestion about Meijer-G definition and identityIntegral with incomplete gamma functionProof of Integral (Gradshteyn & Ryzhik 3.462.1)Challenging definite integral of hypergeometric functionsDefinite integral of exponential, power and Bessel function : a hypergeometric function?Double Gaussian definite integral with one variable limitQuestion over limit of integrand for definite integral.Sum involving hypergeometric 2F2 functionBounding a polynomial from below
$begingroup$
This question is related to Bounding a polynomial from below but perhaps more straightforward.
Let $sigma >0$ be fixed. For even $k in mathbbN cup 0$, we consider the hypergeometric function
beginequation
varphi_k(x) = _2F_1(-k,k+sigma+frac12;frac12;x^2), quad x in (-1,1).
endequation
My question is the following:
Can we find a "closed form" for the definite integral
beginequation
int_-1^1 (1-x^2)^sigma varphi_k(x)^2 dx
endequation
for all even $kin mathbbN cup 0$?
Some remarks:
Since $k$ is an integer, $varphi_k$ terminates and is thus a polynomial of the form
beginequation
varphi_k(x) = sum_j=0^k (-1)^j k choose j b_j , x^2j quad x in (-1,1),
endequation
where
$b_j = fracBig(k+sigma+frac12Big)_jBig(frac12 Big)_j$
and for $s in mathbbR$, $(s)_j$ denotes the Pochhammer symbol
beginequation
(s)_j=begincases1&j=0\s(s+1)cdots (s+j-1)&j>0.endcases
endequation
In particular, $varphi_0(x) =1$.
Writing the square of the sum as the double sum of the product of the coefficients, and using the linearity of the integral, I obtain something of the form
beginequation
sum_j=0^k sum_ell = 0^k (-1)^j+ell k choose ell k choose j fracb_j b_ell2(j+ell)+1 int_-1^1 (1-x^2)^sigma x^2(ell+j) dx.
endequation
Moreover, according to Wolfram Alpha,
beginequation
int_-1^1 (1-x^2)^sigma x^2(ell+j) dx = fracGamma(j+ell+frac12)Gamma(sigma+1)Gamma(sigma+ell+j+frac32).
endequation
I am interested however in knowing if this can be reduced to an expression which does not contain a double sum and is more "compact".
I made this remark in the previous question, and perhaps they might be useful.
polynomials definite-integrals special-functions
asked Mar 27 at 20:01
bgskbgsk
190111
$endgroup$
add a comment |
$begingroup$
This question is related to Bounding a polynomial from below but perhaps more straightforward.
Let $sigma >0$ be fixed. For even $k in mathbbN cup 0$, we consider the hypergeometric function
beginequation
varphi_k(x) = _2F_1(-k,k+sigma+frac12;frac12;x^2), quad x in (-1,1).
endequation
My question is the following:
Can we find a "closed form" for the definite integral
beginequation
int_-1^1 (1-x^2)^sigma varphi_k(x)^2 dx
endequation
for all even $kin mathbbN cup 0$?
Some remarks:
Since $k$ is an integer, $varphi_k$ terminates and is thus a polynomial of the form
beginequation
varphi_k(x) = sum_j=0^k (-1)^j k choose j b_j , x^2j quad x in (-1,1),
endequation
where
$b_j = fracBig(k+sigma+frac12Big)_jBig(frac12 Big)_j$
and for $s in mathbbR$, $(s)_j$ denotes the Pochhammer symbol
beginequation
(s)_j=begincases1&j=0\s(s+1)cdots (s+j-1)&j>0.endcases
endequation
In particular, $varphi_0(x) =1$.
Writing the square of the sum as the double sum of the product of the coefficients, and using the linearity of the integral, I obtain something of the form
beginequation
sum_j=0^k sum_ell = 0^k (-1)^j+ell k choose ell k choose j fracb_j b_ell2(j+ell)+1 int_-1^1 (1-x^2)^sigma x^2(ell+j) dx.
endequation
Moreover, according to Wolfram Alpha,
beginequation
int_-1^1 (1-x^2)^sigma x^2(ell+j) dx = fracGamma(j+ell+frac12)Gamma(sigma+1)Gamma(sigma+ell+j+frac32).
endequation
I am interested however in knowing if this can be reduced to an expression which does not contain a double sum and is more "compact".
I made this remark in the previous question, and perhaps they might be useful.
polynomials definite-integrals special-functions
asked Mar 27 at 20:01
bgskbgsk
190111
$endgroup$
1
$begingroup$
I don't have a solution, but I'd note that $$ (1-x^2)^sigma phi_k(x) = ,_2F_1(-k-sigma,k+frac12;frac12;x^2)$$ That may be useful.
$endgroup$
– Adam Latosiński
Mar 28 at 0:21
add a comment |
$begingroup$
This question is related to Bounding a polynomial from below but perhaps more straightforward.
Let $sigma >0$ be fixed. For even $k in mathbbN cup 0$, we consider the hypergeometric function
beginequation
varphi_k(x) = _2F_1(-k,k+sigma+frac12;frac12;x^2), quad x in (-1,1).
endequation
My question is the following:
Can we find a "closed form" for the definite integral
beginequation
int_-1^1 (1-x^2)^sigma varphi_k(x)^2 dx
endequation
for all even $kin mathbbN cup 0$?
Some remarks:
Since $k$ is an integer, $varphi_k$ terminates and is thus a polynomial of the form
beginequation
varphi_k(x) = sum_j=0^k (-1)^j k choose j b_j , x^2j quad x in (-1,1),
endequation
where
$b_j = fracBig(k+sigma+frac12Big)_jBig(frac12 Big)_j$
and for $s in mathbbR$, $(s)_j$ denotes the Pochhammer symbol
beginequation
(s)_j=begincases1&j=0\s(s+1)cdots (s+j-1)&j>0.endcases
endequation
In particular, $varphi_0(x) =1$.
Writing the square of the sum as the double sum of the product of the coefficients, and using the linearity of the integral, I obtain something of the form
beginequation
sum_j=0^k sum_ell = 0^k (-1)^j+ell k choose ell k choose j fracb_j b_ell2(j+ell)+1 int_-1^1 (1-x^2)^sigma x^2(ell+j) dx.
endequation
Moreover, according to Wolfram Alpha,
beginequation
int_-1^1 (1-x^2)^sigma x^2(ell+j) dx = fracGamma(j+ell+frac12)Gamma(sigma+1)Gamma(sigma+ell+j+frac32).
endequation
I am interested however in knowing if this can be reduced to an expression which does not contain a double sum and is more "compact".
I made this remark in the previous question, and perhaps they might be useful.
polynomials definite-integrals special-functions
asked Mar 27 at 20:01
bgskbgsk
190111
$endgroup$
This question is related to Bounding a polynomial from below but perhaps more straightforward.
Let $sigma >0$ be fixed. For even $k in mathbbN cup 0$, we consider the hypergeometric function
beginequation
varphi_k(x) = _2F_1(-k,k+sigma+frac12;frac12;x^2), quad x in (-1,1).
endequation
My question is the following:
Can we find a "closed form" for the definite integral
beginequation
int_-1^1 (1-x^2)^sigma varphi_k(x)^2 dx
endequation
for all even $kin mathbbN cup 0$?
Some remarks:
Since $k$ is an integer, $varphi_k$ terminates and is thus a polynomial of the form
beginequation
varphi_k(x) = sum_j=0^k (-1)^j k choose j b_j , x^2j quad x in (-1,1),
endequation
where
$b_j = fracBig(k+sigma+frac12Big)_jBig(frac12 Big)_j$
and for $s in mathbbR$, $(s)_j$ denotes the Pochhammer symbol
beginequation
(s)_j=begincases1&j=0\s(s+1)cdots (s+j-1)&j>0.endcases
endequation
In particular, $varphi_0(x) =1$.
Writing the square of the sum as the double sum of the product of the coefficients, and using the linearity of the integral, I obtain something of the form
beginequation
sum_j=0^k sum_ell = 0^k (-1)^j+ell k choose ell k choose j fracb_j b_ell2(j+ell)+1 int_-1^1 (1-x^2)^sigma x^2(ell+j) dx.
endequation
Moreover, according to Wolfram Alpha,
beginequation
int_-1^1 (1-x^2)^sigma x^2(ell+j) dx = fracGamma(j+ell+frac12)Gamma(sigma+1)Gamma(sigma+ell+j+frac32).
endequation
I am interested however in knowing if this can be reduced to an expression which does not contain a double sum and is more "compact".
I made this remark in the previous question, and perhaps they might be useful.
polynomials definite-integrals special-functions
polynomials definite-integrals special-functions
asked Mar 27 at 20:01
bgskbgsk
190111
asked Mar 27 at 20:01
bgskbgsk
190111
edited Mar 27 at 20:29
bgsk
asked Mar 27 at 20:01
bgskbgsk
190111
asked Mar 27 at 20:01
bgskbgsk
190111
asked Mar 27 at 20:01
bgskbgsk
190111
190111
1
$begingroup$
I don't have a solution, but I'd note that $$ (1-x^2)^sigma phi_k(x) = ,_2F_1(-k-sigma,k+frac12;frac12;x^2)$$ That may be useful.
$endgroup$
– Adam Latosiński
Mar 28 at 0:21
add a comment |
1
$begingroup$
I don't have a solution, but I'd note that $$ (1-x^2)^sigma phi_k(x) = ,_2F_1(-k-sigma,k+frac12;frac12;x^2)$$ That may be useful.
$endgroup$
– Adam Latosiński
Mar 28 at 0:21
1
1
$begingroup$
I don't have a solution, but I'd note that $$ (1-x^2)^sigma phi_k(x) = ,_2F_1(-k-sigma,k+frac12;frac12;x^2)$$ That may be useful.
$endgroup$
– Adam Latosiński
Mar 28 at 0:21
$begingroup$
I don't have a solution, but I'd note that $$ (1-x^2)^sigma phi_k(x) = ,_2F_1(-k-sigma,k+frac12;frac12;x^2)$$ That may be useful.
$endgroup$
– Adam Latosiński
Mar 28 at 0:21
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3165055%2fdefinite-integral-of-a-hypergeometric-function%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3165055%2fdefinite-integral-of-a-hypergeometric-function%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
1
$begingroup$
I don't have a solution, but I'd note that $$ (1-x^2)^sigma phi_k(x) = ,_2F_1(-k-sigma,k+frac12;frac12;x^2)$$ That may be useful.
$endgroup$
– Adam Latosiński
Mar 28 at 0:21