Shifting roots in infinite sums of polynomials
Define
$$P(n)=(n-r_1)(n-r_2)...(n-r_k)$$
where $r_iinmathbb{Q}$ and $r_ineq r_j$ for $ineq j$. Now, define
$$Q(n)=(n-(r_1+m))(n-r_2)...(n-r_k)$$
where $minmathbb{Z}$ and $r_1+mneq r_j$ for $1<jleq k$. If we know
$$sum_{n=1}^{infty}frac{1}{P(n)}inmathbb{Q}$$
and $r_1+mnotinmathbb{N}$, does that imply
$$sum_{n=1}^{infty}frac{1}{Q(n)}inmathbb{Q}?$$
For example, it can be shown that
$$sum_{n=1}^{infty}frac{1}{(n-frac{1}{2})(n-frac{5}{2})}=-frac{4}{3} text{ while } sum_{n=1}^{infty}frac{1}{(n-(frac{1}{2}+1))(n-frac{5}{2})}=-frac{3}{2}.$$
The motivation for this problem is that I found a link between my last question on this site and this problem. That is, if one could prove this, then they would prove the other question (the easier one about simple roots). Overall, I have tried to work through this problem using residues and generating functions. Any tips, terms, papers, methods, or generally topics that I could look into would also be welcome.
real-analysis generating-functions residue-calculus rationality-testing
asked 2 days ago
Nick GuerreroNick Guerrero
537411
add a comment |
Define
$$P(n)=(n-r_1)(n-r_2)...(n-r_k)$$
where $r_iinmathbb{Q}$ and $r_ineq r_j$ for $ineq j$. Now, define
$$Q(n)=(n-(r_1+m))(n-r_2)...(n-r_k)$$
where $minmathbb{Z}$ and $r_1+mneq r_j$ for $1<jleq k$. If we know
$$sum_{n=1}^{infty}frac{1}{P(n)}inmathbb{Q}$$
and $r_1+mnotinmathbb{N}$, does that imply
$$sum_{n=1}^{infty}frac{1}{Q(n)}inmathbb{Q}?$$
For example, it can be shown that
$$sum_{n=1}^{infty}frac{1}{(n-frac{1}{2})(n-frac{5}{2})}=-frac{4}{3} text{ while } sum_{n=1}^{infty}frac{1}{(n-(frac{1}{2}+1))(n-frac{5}{2})}=-frac{3}{2}.$$
The motivation for this problem is that I found a link between my last question on this site and this problem. That is, if one could prove this, then they would prove the other question (the easier one about simple roots). Overall, I have tried to work through this problem using residues and generating functions. Any tips, terms, papers, methods, or generally topics that I could look into would also be welcome.
real-analysis generating-functions residue-calculus rationality-testing
asked 2 days ago
Nick GuerreroNick Guerrero
537411
add a comment |
Define
$$P(n)=(n-r_1)(n-r_2)...(n-r_k)$$
where $r_iinmathbb{Q}$ and $r_ineq r_j$ for $ineq j$. Now, define
$$Q(n)=(n-(r_1+m))(n-r_2)...(n-r_k)$$
where $minmathbb{Z}$ and $r_1+mneq r_j$ for $1<jleq k$. If we know
$$sum_{n=1}^{infty}frac{1}{P(n)}inmathbb{Q}$$
and $r_1+mnotinmathbb{N}$, does that imply
$$sum_{n=1}^{infty}frac{1}{Q(n)}inmathbb{Q}?$$
For example, it can be shown that
$$sum_{n=1}^{infty}frac{1}{(n-frac{1}{2})(n-frac{5}{2})}=-frac{4}{3} text{ while } sum_{n=1}^{infty}frac{1}{(n-(frac{1}{2}+1))(n-frac{5}{2})}=-frac{3}{2}.$$
The motivation for this problem is that I found a link between my last question on this site and this problem. That is, if one could prove this, then they would prove the other question (the easier one about simple roots). Overall, I have tried to work through this problem using residues and generating functions. Any tips, terms, papers, methods, or generally topics that I could look into would also be welcome.
real-analysis generating-functions residue-calculus rationality-testing
asked 2 days ago
Nick GuerreroNick Guerrero
537411
Define
$$P(n)=(n-r_1)(n-r_2)...(n-r_k)$$
where $r_iinmathbb{Q}$ and $r_ineq r_j$ for $ineq j$. Now, define
$$Q(n)=(n-(r_1+m))(n-r_2)...(n-r_k)$$
where $minmathbb{Z}$ and $r_1+mneq r_j$ for $1<jleq k$. If we know
$$sum_{n=1}^{infty}frac{1}{P(n)}inmathbb{Q}$$
and $r_1+mnotinmathbb{N}$, does that imply
$$sum_{n=1}^{infty}frac{1}{Q(n)}inmathbb{Q}?$$
For example, it can be shown that
$$sum_{n=1}^{infty}frac{1}{(n-frac{1}{2})(n-frac{5}{2})}=-frac{4}{3} text{ while } sum_{n=1}^{infty}frac{1}{(n-(frac{1}{2}+1))(n-frac{5}{2})}=-frac{3}{2}.$$
The motivation for this problem is that I found a link between my last question on this site and this problem. That is, if one could prove this, then they would prove the other question (the easier one about simple roots). Overall, I have tried to work through this problem using residues and generating functions. Any tips, terms, papers, methods, or generally topics that I could look into would also be welcome.
real-analysis generating-functions residue-calculus rationality-testing
real-analysis generating-functions residue-calculus rationality-testing
asked 2 days ago
Nick GuerreroNick Guerrero
537411
asked 2 days ago
Nick GuerreroNick Guerrero
537411
asked 2 days ago
Nick GuerreroNick Guerrero
537411
asked 2 days ago
Nick GuerreroNick Guerrero
537411
asked 2 days ago
Nick GuerreroNick Guerrero
537411
537411
add a comment |
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
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%2f3063141%2fshifting-roots-in-infinite-sums-of-polynomials%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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3063141%2fshifting-roots-in-infinite-sums-of-polynomials%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
