Category : Mathematics – Calculations

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

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

(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 t[...]
[Calculation 19] Even values of the Zeta Function

[Calculation 19] Even values of the Zeta Function

(revision of 2012)  In this post, we evaluate even values of the Riemann zeta function. Calculation 19. For $nin N$, we have $$ zeta(2n) = frac{|B_[...]
[Calculation 18] Some Useful Formulas from the Jacobi Triple Product

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

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, wh[...]
[Calculation 17] Jacobi Triple Product Formula

[Calculation 17] Jacobi Triple Product Formula

We introduce the Jacobi Triple Product Formula [1] here. Actually it can be easily obtained from the Ramanujan ${}_1psi_1$ Summation Formula [2] with some proper coeffici[...]
[Calculation 16] Ramanujan 's 1ψ1 (1-psi-1) Summation Formula

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

In this post, we will introduce one of the famous formulas discovered by Ramanujan, which is called Ramanujan's ${}_1psi_1$ Summation Formula. It was first introduced by [...]
[Calculation 15] Introduction to the Ramanujan Theta Functions

[Calculation 15] Introduction to the Ramanujan Theta Functions

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, a[...]
[Calculation 14] q-Series and the q-Binomial Theorem

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

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|[...]
[Calculation 13] A Simple Formula Related to Digamma Functions

[Calculation 13] A Simple Formula Related to Digamma Functions

Theorem. The following holds: $$ pi ;{}_2 F_1 left( frac{1}{2}, frac{1}{2}; 1; 1-xright) = log left( frac{16}{x}right) {}_2 F_1 left( frac{1}{2}, frac{1}{2};1;[...]
[Calculation 12] A Basic Formula for Hypergeometric Functions

[Calculation 12] A Basic Formula for Hypergeometric Functions

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-zright)\ &, qquadqquadqquad=frac{Gamma(a+b+n+2)Gamma[...]
[Calculation 11] Simple Examples from the Dixon Theorem

[Calculation 11] Simple Examples from the Dixon Theorem

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[...]
[Calculation 10] Dixon Theorem

[Calculation 10] Dixon Theorem

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 be[...]
[Calculation 9] Simple Corollaries from Gauss and Bailey Formula

[Calculation 9] Simple Corollaries from Gauss and Bailey Formula

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-zri[...]
[Calculation 8] Gauss's Quadratic Trasformation

[Calculation 8] Gauss's Quadratic Trasformation

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[...]
[Calculation 7] Bailey's Formulas for Hypergeometric Functions

[Calculation 7] Bailey's Formulas for Hypergeometric Functions

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|<[...]
[Calculation 6] Kummer's Theorem

[Calculation 6] Kummer's Theorem

Theorem. (Kummer's Theorem) $$ {}_2 F_1 (a,b;1+a-b;-1) = frac{Gamma(1+a-b)Gammaleft(1+frac{1}{2}aright)}{Gamma(1+a)Gammaleft(1+frac{1}{2}a-bright)} $$ [...]
[Calculation 5] Gauss's Summation Formula

[Calculation 5] Gauss's Summation Formula

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. [...]
[Calculation 4] Saalschütz's Theorem

[Calculation 4] Saalschütz's Theorem

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 $nin mathbb N_0$.   [...]
[Calculation 3] Euler's Transformation Formula

[Calculation 3] Euler's Transformation Formula

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 [...]
[Calculation 2] Pfaff’s Transformation Formula

[Calculation 2] Pfaff’s Transformation Formula

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[...]
[Calculation 1] Fundamentals of Hypergeometric Functions

[Calculation 1] Fundamentals of Hypergeometric Functions

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[...]