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Calculations / Mathematics

[Calculation 20] Simple Series Calculation for Σ1/(n²ᵏ+c)

Posted On March 6, 2014

(revision of 2012) In this post, we will evaluate the closed form of the series $\sum_n (n^{2k}+x^k)^{-1}$ for $k=2,4,6,\cdots$ and $x>0$. We are going to show the following formula. Calculation 20. For …

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Calculations / Mathematics

[Calculation 19] Even values of the Zeta Function

March 4, 2014

(revision of 2012) In this post, we evaluate even values of the Riemann zeta function. Calculation 19. For $n\in \N$, we have $$ \zeta(2n) = \frac{|B_{2n}|2^{2n-1} \pi^{2n}}{(2n)!}, $$ where $B_n$'s are Bernoulli numbers. Proof. We have the …

Calculations / Mathematics

[Calculation 18] Some Useful Formulas from the Jacobi Triple Product

December 4, 2013

In this post, we introduce some useful formulas, which can be proved by the Jacobi Triple Product [2]. As before, we always denote $f$ as the Ramanujan Theta Function, which …

Calculations / Mathematics

[Calculation 17] Jacobi Triple Product Formula

December 3, 2013

We introduce the Jacobi Triple Product Formula [1] here. Actually it can be easily obtained from the Ramanujan ${}_1\psi_1$ Summation Formula [2] with some proper coefficients. Here we denote $f$ …

Calculations / Mathematics

[Calculation 16] Ramanujan ‘s 1ψ1 (1-psi-1) Summation Formula

December 2, 2013

In this post, we will introduce one of the famous formulas discovered by Ramanujan, which is called Ramanujan's ${}_1\psi_1$ Summation Formula. It was first introduced by Hardy, and he called …

Calculations / Mathematics

[Calculation 15] Introduction to the Ramanujan Theta Functions

November 30, 2013

In this post, we introduce the Ramanujan theta functions $f(a,b)$, which generalize the form of the Jacobi theta functions. Here we define the Ramanujan theta function, and introduce some elementary …

Calculations / Mathematics

[Calculation 14] q-Series and the q-Binomial Theorem

November 30, 2013

In this post, we introduce q-Series and the q-Binomial theorem. For any complex number $a$, we write $$ (a;q)_k = (1-a)(1-aq)(1-aq^2) \cdots (1-aq^{k-1}) $$ where $|q|<1$. Also we write $$ (a;q)_\infty = \prod_{k=0}^\infty (1-aq^k). $$ With these notations, …

Calculations / Mathematics

[Calculation 13] A Simple Formula Related to Digamma Functions

November 29, 2013

Theorem. The following holds: $$ \pi \;{}_2 F_1 \left( \frac{1}{2}, \frac{1}{2}; 1; 1-x\right) = \log \left( \frac{16}{x}\right) {}_2 F_1 \left( \frac{1}{2}, \frac{1}{2};1;x\right) - 4 \sum_{k=1}^\infty \frac{(\frac{1}{2})_k^2}{(k!)^2} \sum_{j=1}^k \frac{x^k}{(2j-1)(2j)}. $$ Proof. First, we recall the Corollary …

Calculations / Mathematics

[Calculation 12] A Basic Formula for Hypergeometric Functions

November 27, 2013

Theorem. Let $n \notin \mathbb Z$. Then we have \begin{align}\tag{1} &{}_2 F_1 \left( a+n+1, b+n+1; a+b+n+2; 1-z\right)\\ &\, \qquad\qquad\qquad=\frac{\Gamma(a+b+n+2)\Gamma(-n)}{\Gamma(a+1)\Gamma(b+1)}\; {}_2 F_1 (a+n+1,b+n+1;n+1;z)\\ &\, \qquad\qquad\qquad\qquad+\frac{\Gamma(a+b+n+2)\Gamma(n)z^{-n}}{\Gamma(a+n+1)\Gamma(b+n+1)} \;{}_2 F_1 (a+1,b+1;-n+1;z). \end{align} Proof. We consider the following ODE, which is called …

Calculations / Mathematics

[Calculation 11] Simple Examples from the Dixon Theorem

November 27, 2013

Here we introduce some examples from the Dixon Theorem. By setting suitable coefficients, we can obtain simple formulas of infinite series, which are related to the Gamma functions. Example. From the …

Calculations / Mathematics

[Calculation 10] Dixon Theorem

November 24, 2013

Here we note Dixon's theorem, which gives some special values of ${}_3 F_2$, since the proof is almost automatic by using Gauss and Kummer's formulas which we've shown before. Theorem. (Dixon's …

Calculations / Mathematics

[Calculation 9] Simple Corollaries from Gauss and Bailey Formula

November 24, 2013

Corollary 1. For $\frac{1}{2} < z < 2$, $$ {}_2 F_1 \left(\frac{1}{2}, \frac{1}{2};1;1-\frac{1}{z}\right) = \sqrt{z} {}_2 F_1 \left(\frac{1}{2}, \frac{1}{2};1;1-z\right). $$ Proof. We recall Bailey's Formula ((i) in [2]) for $w\in\mathbb R$: \begin{equation}\tag{1} (1-w)^{-a} {}_2 F_1 \left( a,b;c; - …

Calculations / Mathematics

[Calculation 8] Gauss’s Quadratic Trasformation

November 23, 2013

Theorem. (Gauss's Quadratic Transformation) \begin{equation}\tag{1} (1+z)^{-2a} {}_2 F_1 \left(a,b; 2b; \frac{4z}{(1+z)^2}\right) = {}_2 F_1 \left(a, 1+\frac{1}{2}-b;b+\frac{1}{2};z^2\right). \end{equation} Proof. The proof is almost similar to that of [2]. Note that the left hand side of (1) can …

Calculations / Mathematics

[Calculation 7] Bailey’s Formulas for Hypergeometric Functions

November 22, 2013

Theorem. (Bailey) The followings are valid: \begin{align*} &\text{(i) } (1-z)^{-a} {}_2 F_1 \left( a,b;c; - \frac{z}{1-z}\right) = {}_2 F_1 (a,c-b;c;z),\quad|z|<1,\;\text{Re}z< \frac{1}{2},\\ &\text{(ii) } {}_2 F_1 \left( a,b; \frac{a+b+1}{2}; \frac{1}{2}\right) = \frac{\Gamma(\frac{1}{2}) \Gamma(\frac{1+a+b}{2})}{\Gamma(\frac{1+a}{2})\Gamma(\frac{1+b}{2})},\\ &\text{(iii) } …

Calculations / Mathematics

[Calculation 6] Kummer’s Theorem

November 22, 2013

Theorem. (Kummer's Theorem) $$ {}_2 F_1 (a,b;1+a-b;-1) = \frac{\Gamma(1+a-b)\Gamma\left(1+\frac{1}{2}a\right)}{\Gamma(1+a)\Gamma\left(1+\frac{1}{2}a-b\right)} $$ To prove Kummer's theorem, we introduce the following lemma, which is called Kummer's quadratic transformation: Lemma. (Kummer's Quadratic Transformation) \begin{equation}\tag{1} {}_2 F_1 (a,b;1+a-b;z) = (1-z)^{-a} {}_2 F_1 …

Calculations / Mathematics

[Calculation 5] Gauss’s Summation Formula

November 22, 2013

Theorem. (Gauss's Summation Formula) For $\text{Re}c>\text{Re}b>0$, $$ {}_2 F_1 (a,b;c;1) = \frac{\Gamma(c) \Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)} $$ holds. Proof. We remember that the Euler Integral Representation for the hypergeometric function is $$ _2 F_1 (a,b;c;z) = \frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)} \int_0^1 \frac{t^{b-1} (1-t)^{c-b-1}}{(1-tz)^a}\,dt. $$ Taking the limit $z\to 1$ …

Calculations / Mathematics

[Calculation 4] Saalschütz’s Theorem

November 22, 2013

Theorem. (Saalschütz's Theorem) $$ {}_3 F_2 (a,b,-n;c,1+a+b-c-n;1) = \frac{(c-a)_n (c-b)_n}{(c)_n (c-a-b)_n} $$ holds for $n\in \mathbb N_0$. Proof. We recall the Euler Transformation Formula: $$(1-z)^{a+b-c} {}_2 F_1 (a,b;c;z) = {}_2 F_1 (c-a,c-b;c;z).$$ Equating the coefficients of $z^n$ …

Calculations / Mathematics

[Calculation 3] Euler’s Transformation Formula

November 22, 2013

Theorem (Euler's Transformation Formula) $${}_2 F_1 (a,b;c;z) = (1-z)^{c-a-b} {}_2 F_1 (c-a,c-b;c;z)$$ Proof. Applying Pfaff’s Transformation Formula twice, we obtain \begin{eqnarray*} {}_2 F_1(a,b;c;z) &=& (1-z)^{-a} {}_2 F_1 \left(a,c-b;c; \frac{z}{z-1} \right)\\ &=& (1-z)^{-a} \left(1-\frac{z}{z-1}\right)^{b-c} {}_2 F_1 \left(c-a,c-b;c;\frac{\frac{z}{z-1}}{\frac{z}{z-1} -1} …

Calculations / Mathematics

[Calculation 2] Pfaff’s Transformation Formula

November 22, 2013

Theorem. (Pfaff’s Transformation Formula) $${}_2 F_1(a,b;c;z) = (1-z)^{-a} {}_2 F_1 \left(a,c-b;c; \frac{z}{z-1} \right)$$ Proof. We remember the Euler Integral Representation for the hypergeometric function: $$ {}_2 F_1 (a,b;c;z) = \frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)} \int_0^1 t^{b-1}(1-t)^{c-b-1} (1-tz)^{-a}\,dt. $$ Substitution $t=1-s$ yields \begin{eqnarray*} {}_2 F_1 …

Calculations / Mathematics

[Calculation 1] Fundamentals of Hypergeometric Functions

November 22, 2013

The classical hypergeometric function ${}_{2}F_1$ is defined by $$ {}_{2}F_1(a,b;c;z) = \sum_{k=0}^\infty \frac{(a)_k(b)_k}{(c)_k} \frac{z^k}{k!} $$ where $(\cdot)_k$ is Pochhammer symbol, that is, $$ (q)_k = \frac{\Gamma(q+k)}{\Gamma(q)} $$ provided that $q+k$ is not a negative integer, with the convention $1/{\pm\infty} = …

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