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fixed computational error
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Dima Pasechnik
Dima Pasechnik
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$S(m,q)$ is a hypergeometric function (you have take the upper limit of the sum $\infty$, as terms for $i$ bigger than $m-q$ will all vanish); a "standard" method would be to find it explicitly, and then to use a representation of it by an integral, which can be estimated by methods from asymptotic analysis.

EDIT: I am referring to the standard technique to identify a hypergeometric series explained in e.g. Chapter 3 of the book "A=B" by Petkovsek, Wilf, and Zeilberger.

Using it you will be able to write $$ S(m,q)=\left(\frac{\binom{m}{q}}{q+1}\right)^2\cdot _3F_0(-m,q-m,q-m;-;-1). $$$$ S(m,q)=\binom{m}{q}^2 \ _3F_0(-m,q-m,q-m;-;\frac{-1}{(q+1)^2}). $$

$S(m,q)$ is a hypergeometric function (you have take the upper limit of the sum $\infty$, as terms for $i$ bigger than $m-q$ will all vanish); a "standard" method would be to find it explicitly, and then to use a representation of it by an integral, which can be estimated by methods from asymptotic analysis.

EDIT: I am referring to the standard technique to identify a hypergeometric series explained in e.g. Chapter 3 of the book "A=B" by Petkovsek, Wilf, and Zeilberger.

Using it you will be able to write $$ S(m,q)=\left(\frac{\binom{m}{q}}{q+1}\right)^2\cdot _3F_0(-m,q-m,q-m;-;-1). $$

$S(m,q)$ is a hypergeometric function (you have take the upper limit of the sum $\infty$, as terms for $i$ bigger than $m-q$ will all vanish); a "standard" method would be to find it explicitly, and then to use a representation of it by an integral, which can be estimated by methods from asymptotic analysis.

EDIT: I am referring to the standard technique to identify a hypergeometric series explained in e.g. Chapter 3 of the book "A=B" by Petkovsek, Wilf, and Zeilberger.

Using it you will be able to write $$ S(m,q)=\binom{m}{q}^2 \ _3F_0(-m,q-m,q-m;-;\frac{-1}{(q+1)^2}). $$

added an explanation
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Dima Pasechnik
Dima Pasechnik
  • 14k
  • 2
  • 34
  • 70

$S(m,q)$ is a hypergeometric function (you have take the upper limit of the sum $\infty$, as terms for $i$ bigger than $m-q$ will all vanish); a "standard" method would be to find it explicitly, and then to use a representation of it by an integral, which can be estimated by methods from asymptotic analysis.

EDIT: I am referring to the standard technique to identify a hypergeometric series explained in e.g. Chapter 3 of the book "A=B" by Petkovsek, Wilf, and Zeilberger.

Using it you will be able to write $$ S(m,q)=\left(\frac{\binom{m}{q}}{q+1}\right)^2\cdot _3F_0(-m,q-m,q-m;-;-1). $$

$S(m,q)$ is a hypergeometric function (you have take the upper limit of the sum $\infty$, as terms for $i$ bigger than $m-q$ will all vanish); a "standard" method would be to find it explicitly, and then to use a representation of it by an integral, which can be estimated by methods from asymptotic analysis.

$S(m,q)$ is a hypergeometric function (you have take the upper limit of the sum $\infty$, as terms for $i$ bigger than $m-q$ will all vanish); a "standard" method would be to find it explicitly, and then to use a representation of it by an integral, which can be estimated by methods from asymptotic analysis.

EDIT: I am referring to the standard technique to identify a hypergeometric series explained in e.g. Chapter 3 of the book "A=B" by Petkovsek, Wilf, and Zeilberger.

Using it you will be able to write $$ S(m,q)=\left(\frac{\binom{m}{q}}{q+1}\right)^2\cdot _3F_0(-m,q-m,q-m;-;-1). $$

Source Link
Dima Pasechnik
Dima Pasechnik
  • 14k
  • 2
  • 34
  • 70

$S(m,q)$ is a hypergeometric function (you have take the upper limit of the sum $\infty$, as terms for $i$ bigger than $m-q$ will all vanish); a "standard" method would be to find it explicitly, and then to use a representation of it by an integral, which can be estimated by methods from asymptotic analysis.