Double series expansions for $ \pi $

Long Li; Long Li

doi:10.3934/math.2021294

AIMS Mathematics

2021, Volume 6, Issue 5: 5000-5007. doi: 10.3934/math.2021294

Previous Article Next Article

Research article

Double series expansions for $\pi$

Long Li ^,

School of Mathematics and Statistics, Huaiyin Normal University, Huai'an 223300, Jiangsu, People's Republic of China

Received: 22 November 2020 Accepted: 18 February 2021 Published: 04 March 2021
MSC : 33C70, 33B15, 65B10

We use some properties of gamma functions and a summation formula for Kampé de Fériet function $F_{1:1;1}^{0:3;3}$ to give many double series expansions for $1/\pi$ and $\pi$ .

Keywords:

Citation: Long Li. Double series expansions for $\pi$ [J]. AIMS Mathematics, 2021, 6(5): 5000-5007. doi: 10.3934/math.2021294

Related Papers:

[1]	Qiuxia Hu, Bilal Khan, Serkan Araci, Mehmet Acikgoz . New double-sum expansions for certain Mock theta functions. AIMS Mathematics, 2022, 7(9): 17225-17235. doi: 10.3934/math.2022948
[2]	Suxia Wang, Tiehong Zhao . New refinements of Becker-Stark inequality. AIMS Mathematics, 2024, 9(7): 19677-19691. doi: 10.3934/math.2024960
[3]	Zhenhua Su, Zikai Tang, Hanyuan Deng . Higher-order Randić index and isomorphism of double starlike trees. AIMS Mathematics, 2023, 8(12): 31186-31197. doi: 10.3934/math.20231596
[4]	Waleed Mohamed Abd-Elhameed, Youssri Hassan Youssri . Spectral tau solution of the linearized time-fractional KdV-Type equations. AIMS Mathematics, 2022, 7(8): 15138-15158. doi: 10.3934/math.2022830
[5]	Tie-Hong Zhao, Zai-Yin He, Yu-Ming Chu . On some refinements for inequalities involving zero-balanced hypergeometric function. AIMS Mathematics, 2020, 5(6): 6479-6495. doi: 10.3934/math.2020418
[6]	Mustafa Inc, Mamun Miah, Akher Chowdhury, Shahadat Ali, Hadi Rezazadeh, Mehmet Ali Akinlar, Yu-Ming Chu . New exact solutions for the Kaup-Kupershmidt equation. AIMS Mathematics, 2020, 5(6): 6726-6738. doi: 10.3934/math.2020432
[7]	Aslıhan ILIKKAN CEYLAN, Canan HAZAR GÜLEÇ . A new double series space derived by factorable matrix and four-dimensional matrix transformations. AIMS Mathematics, 2024, 9(11): 30922-30938. doi: 10.3934/math.20241492
[8]	Waleed Mohamed Abd-Elhameed, Abdullah F. Abu Sunayh, Mohammed H. Alharbi, Ahmed Gamal Atta . Spectral tau technique via Lucas polynomials for the time-fractional diffusion equation. AIMS Mathematics, 2024, 9(12): 34567-34587. doi: 10.3934/math.20241646
[9]	Ling Zhu . New inequalities of Wilker's type for hyperbolic functions. AIMS Mathematics, 2020, 5(1): 376-384. doi: 10.3934/math.2020025
[10]	Bai-Ni Guo, Dongkyu Lim, Feng Qi . Series expansions of powers of arcsine, closed forms for special values of Bell polynomials, and series representations of generalized logsine functions. AIMS Mathematics, 2021, 6(7): 7494-7517. doi: 10.3934/math.2021438

Abstract

We use some properties of gamma functions and a summation formula for Kampé de Fériet function $F_{1:1;1}^{0:3;3}$ to give many double series expansions for $1/\pi$ and $\pi$ .

1. Introduction

In ^[18] Ramanujan showed a total of 17 series for $1/\pi$ but he did not indicate how he arrived at these series. The Borwein brothers ^[5] gave rigorous proofs of Ramanujan's series for the first time and also obtained many new series for $1/\pi.$ Till now, many new Ramanujan's-type series for $1/\pi$ have been published, (see, for example, ^[4,6,8]). Chu ^[7], Liu ^[15,16] and Wei et al. ^[21,22] gave many $\pi$ -formula with free parameters by means of gamma functions and hypergeometric series. Guillera ^[10] proved a kind of bilateral semi-terminating series related to Ramanujan-like series for negative powers of $\pi$ . Moreover, Guillera and Zudilin ^[11] outlined an elementary method for proving numerical hypergeometric identities, in particular, Ramanujan-type identities for $1/\pi$ . Recently, $q$ -analogues of Ramanujan-type series for $1/\pi$ have caught the interests of many authors (see, for example, ^{[9,12,13,14,20,21]}).

Although various definitions for gamma functions are used in the literature, we adopt the following definition [23, p.76]

$\frac 1{\Gamma(z)} = ze^{\gamma z}\prod\limits_{n = 1}^\infty\left(1+\frac zn\right)e^{-\frac zn}$

where $\gamma$ is the Euler constant defined as

$\gamma = \lim\limits_{n\rightarrow \infty}\left(1+ \frac 12+\cdots+ \frac 1n-\log n\right).$

It is easy to verify that $\Gamma(1) = 1, \Gamma({ \frac 12}) = \sqrt\pi$ and $\Gamma({z+1}) = z\Gamma(z).$ It follows that for every positive integer $n$ , $\Gamma(n) = (n-1)!.$

For any complex $\alpha$ , we define the general rising shifted factorial by

$\begin{align} (z)_{{\alpha}} = \Gamma({z+{\alpha}})/\Gamma(z). \end{align}$

(1.1)

Obviously, $(z)_0 = 1.$ For every positive integer $n$ , we have

$(z)_n = \Gamma({z+n})/\Gamma(z) = z(z+1)\cdots(z+n-1)$

and

$(z)_{-n} = \Gamma({z-n})/\Gamma(z) = \frac 1{(z-1)(z-2)\ldots(z-n)}.$

For convenience, we use the following compact notations

$(a_1,a_2,\ldots,a_m)_n = (a_1)_n(a_2)_n\ldots(a_m)_n$

and

$(a)_{(n_1,n_2,\ldots,n_m)} = (a)_{n_1}(a)_{n_2}\ldots(a)_{n_m}.$

Following ^[1,3], the hypergeometric series is defined by

$_{{r+1}}F_{s}\left[\begin{array}{cccc}a_0,&a_1,&\ldots,&a_r\\&b_1,&\ldots,&b_s\end{array};z\right] = \sum\limits_{k = 0}^\infty \frac {(a_0, a_1,\ldots,a_r)_k}{(b_1,\ldots,b_s)_k} \frac {z^k}{k!},$

where $a_i, b_j(i = 0, 1, \ldots, r, j = 1, 2, \ldots, s)$ are complex numbers such that no zero factors appear in the denominators of the summand on the right hand side.

We let $F_{q:s; v}^{p:r; u}\ ({p, q, r, s, u, v\in \Bbb N_0 = \{0, 1, 2, \ldots\}})$ denote a general (Kampé de Fériet's) double hypergeometric function defined by (see ^[2,19])

$\begin{eqnarray*} &&F_{q:s;v}^{p:r;u}\left[\begin{array}{ccc}{\alpha}_1,\ldots,{\alpha}_p:&a_1,\ldots,a_r;&c_1,\ldots,c_u;\\ \beta_1,\ldots,\beta_q:&b_1,\ldots,b_s;&d_1,\ldots,d_v;\end{array}x,y\right]\\&& = \sum\limits_{m,n = 0}^\infty \frac {({\alpha}_1,\ldots,{\alpha}_p)_{m+n} (a_1,\ldots,a_r)_m(c_1,\ldots,c_u)_n}{(\beta_1,\ldots,\beta_q)_{m+n} (b_1,\ldots,b_s)_m(d_1,\ldots,d_v)_n} \frac {x^m}{m!} \frac {y^n}{n!}, \end{eqnarray*}$

where, for convergence of the double hypergeometric series,

$p+r\leq q+s+1{\quad}\text{and}{\quad} p+u\leq q+v+1,$

with equality only when $|x|$ and $|y|$ are appropriately constrained (see, for details, [19,Eq 1.3(29),p.27]).

There exist numerous identities for such series. For example, we have

Theorem 1.1 (See [,(30)] ) If $Re(e-d) > 0$ and $Re(d+e-a-b-c) > 0,$ then

$F_{1:1;1}^{0:3;3}\left[\begin{array}{ccc}-:&a,b,c;&d-a,d-b,d-c;\\d:&e;&d+e-a-b-c;\end{array}1,1\right] = \frac {\Gamma({e})\Gamma({e+d-a-b-c})\Gamma({e-d})}{\Gamma({e-a})\Gamma({e-b})\Gamma({e-c})}.$

In ^[15], Liu used the general rising shifted factorial and the Gauss summation formula to prove the following four-parameter series expansions formula, which implies infinitely many Ramanujan type series for $1/\pi$ and $\pi$ .

Theorem 1.2 For any complex ${\alpha}$ and $Re(c-a-b) > 0,$ we have

$\sum\limits_{n = 0}^\infty \frac {({\alpha})_{a+n}(1-{\alpha})_{b+n}}{n!\Gamma({c+n+1})} = \frac {({\alpha})_a(1-{\alpha})_b\Gamma({c-a-b})} {({\alpha})_{c-b}(1-{\alpha})_{c-a}}\cdot \frac {\sin \pi{\alpha}}\pi.$

Motivated by Liu's work, in this paper we derive the following result from Theorem 1.1 which enables us to give many double series expansions for $1/\pi$ and $\ \pi$ . To the best of our knowledge, most of the results in this paper have not previously appeared.

Theorem 1.3 If $d\in \Bbb N_0, Re(e-d+{\sigma}-{\delta}) > 0$ and $Re(d+e-a-b-c+{\delta}+{\sigma}-{\alpha}-\beta-{\gamma}) > 0,$ then

$\begin{align*} &\sum\limits_{m,n = 0}^\infty \frac {({\alpha})_{a+m}(\beta)_{b+m}({\gamma})_{c+m}({\delta}-{\alpha})_{d-a+n}({\delta}-\beta)_{d-b+n}({\delta}-{\gamma})_{d-c+n}} {m!n!({\delta}+d)_{m+n}({\sigma})_{e+m}({\delta}+{\sigma}-{\alpha}-\beta-{\gamma})_{d+e-a-b-c+n}}\\ & = \frac {({\alpha})_a(\beta)_b({\gamma})_c({\delta}-{\alpha})_{d-a}({\delta}-\beta)_{d-b}({\delta}-{\gamma})_{d-c}({\sigma}-{\delta})_{e-d}} {({\sigma}-{\alpha})_{e-a}({\sigma}-\beta)_{e-b}({\sigma}-{\gamma})_{e-c}} \cdot \frac {\Gamma({\sigma})\Gamma({{\sigma}-{\delta}})\Gamma({{\delta}+{\sigma}-{\alpha}-\beta-{\gamma}})}{\Gamma({{\sigma}-{\alpha}})\Gamma({{\sigma}-\beta})\Gamma({{\sigma}-{\gamma}})}. \end{align*}$

Several examples of such formulae are

$\sum\limits_{m,n = 0}^\infty \frac {( \frac 12)_m^3( \frac 12)_n^2}{m!n!(m+n)!(m+1)!(2n+1)} = \frac 4{\pi},$

$\sum\limits_{m,n = 0}^\infty \frac {(- \frac 12)_m^3( \frac 32)_n^3}{m!n!(m+n)!(n+3)!( \frac 12)_{m+1}} = \pi,$

and

$\sum\limits_{m,n = 0}^\infty \frac {(- \frac 23)_m^2( \frac 13)_n^3}{m!n!(n+1)!(2-3m)(- \frac 13)_{m+n}} = \frac {\sqrt3\pi}3.$

The remainder of the paper is organized as follows. In section 2 we give the proof of Theorem 1.3. Sections 3 and 4 are devoted to the double series expansions for $1/\pi$ and $\pi$ , respectively.

2. Proof of Theorem 1.3

First of all, by making use of (1.1), Theorem 1.3 can be restated as follows:

$\begin{equation} \begin{split} &\sum\limits_{m,n = 0}^\infty \frac {\Gamma({a+m})\Gamma({b+m})\Gamma({c+m})\Gamma({d-a+n})\Gamma({d-b+n})\Gamma({d-c+n})} {m!n!\Gamma({d+m+n})\Gamma({e+m})\Gamma({d+e-a-b-c+n})}\\ & = \frac {\Gamma(a)\Gamma(b)\Gamma(c)\Gamma({d-a})\Gamma({d-b})\Gamma({d-c})\Gamma({e-d})}{\Gamma(d)\Gamma({e-a})\Gamma({e-b})\Gamma({e-c})}. \end{split} \end{equation}$

(2.1)

From (1.1) it is easy to see that

$\begin{align*} &\Gamma({a+{\alpha}+m}) = ({\alpha})_{a+m}\Gamma({{\alpha}}), \ \Gamma({b+\beta+m}) = (\beta)_{b+m}\Gamma({\beta}),\ \Gamma({c+{\gamma}+m}) = ({\gamma})_{c+m}\Gamma({{\gamma}}),\\ &\Gamma({d-a+{\delta}-{\alpha}+n}) = ({\delta}-{\alpha})_{d-a+n}\Gamma({{\delta}-{\alpha}}),\ \Gamma({d-b+{\delta}-\beta+n}) = ({\delta}-\beta)_{d-b+n}\Gamma({{\delta}-\beta}),\\ &\Gamma({d-c+{\delta}-{\gamma}+n}) = ({\delta}-{\gamma})_{d-c+n}\Gamma({{\delta}-{\gamma}}),\ \Gamma({d+{\delta}+m+n}) = ({\delta})_{d+m+n}\Gamma({{\delta}})\\ &\Gamma({e+m+{\sigma}}) = ({\sigma})_{e+m}\Gamma({\sigma}),\ \Gamma({a+{\alpha}}) = ({\alpha})_a\Gamma({\alpha}),\ \Gamma({b+\beta}) = (\beta)_b\Gamma(\beta),\ \Gamma({c+{\gamma}}) = ({\gamma})_c\Gamma({\gamma}),\\ &\Gamma({d-a+{\delta}-{\alpha}}) = ({\delta}-{\alpha})_{d-a}\Gamma({{\delta}-{\alpha}}),\ \Gamma({d-b+{\delta}-\beta}) = ({\delta}-\beta)_{d-b}\Gamma({{\delta}-\beta}),\\ &\Gamma({d-c+{\delta}-{\gamma}}) = ({\delta}-{\gamma})_{d-c}\Gamma({{\delta}-{\gamma}}),\ \Gamma({e-d+{\sigma}-{\delta}}) = ({\sigma}-{\delta})_{e-d}\Gamma({{\sigma}-{\delta}}),\\ &\Gamma({d+{\delta}}) = ({\delta})_d\Gamma({\delta}),{\quad} \Gamma({e-a+{\sigma}-{\alpha}}) = ({\sigma}-{\alpha})_{e-a}\Gamma({{\sigma}-{\alpha}}),\\ &\Gamma({e-b+{\sigma}-\beta}) = ({\sigma}-\beta)_{e-b}\Gamma({{\sigma}-\beta}),\ \Gamma({e-c+{\sigma}-{\gamma}}) = ({\sigma}-{\gamma})_{e-c}\Gamma({{\sigma}-{\gamma}}),\\ &\Gamma({d+e-a-b-c+{\delta}+{\sigma}-{\alpha}-\beta-{\gamma}})\\& = ({\delta}+{\sigma}-{\alpha}-\beta-{\gamma})_{d+e-a-b-c}\Gamma({{\delta}+{\sigma}-{\alpha}-\beta-{\gamma}}). \end{align*}$

and we realize that $({\delta})_{d+m+n} = ({\delta})_d({\delta}+d)_{m+n}$ when $d\in \Bbb N_0.$ Replacing $(a, b, c, d, e)$ by $(a+{\alpha}, b+\beta, c+{\gamma}, d+{\delta}, e+{\sigma})$ in (2.1) and substituting above identities into the resulting equation, we get the desired result.

3. Double series expansions for $1/\pi$

In this section we will use Theorem 1.3 to prove the following double series expansion formula for $1/\pi$ .

Theorem 3.1 If $d\in \Bbb N_0, Re(e-d+1) > 0$ and $Re(d+e-a-b-c+ \frac 32) > 0,$ then

$\sum\limits_{m,n = 0}^\infty \frac {( \frac 12)_{(a+m,b+m,c+m,d-a+n,d-b+n,d-c+n)}} {m!n!(d+1)_{m+n}(2)_{e+m}( \frac 32)_{d+e-a-b-c+n}} = \frac {( \frac 12)_{(a,b,c,d-a,d-b,d-c)}(1)_{e-d}} {( \frac 32)_{(e-a,e-b,e-c)}}\cdot \frac 4{\pi}.$

Proof. Let $({\alpha}, \beta, {\gamma}, {\delta}, {\sigma}) = ( \frac 12, \frac 12, \frac 12, 1, 2)$ in Theorem 1.3. We find that

$\begin{align} \sum\limits_{m,n = 0}^\infty \frac {( \frac 12)_{(a+m,b+m,c+m,d-a+n,d-b+n,d-c+n)}} {m!n!(d+1)_{m+n}(2)_{e+m}( \frac 32)_{d+e-a-b-c+n}} = \frac {( \frac 12)_{(a,b,c,d-a,d-b,d-c)}(1)_{e-d}} {( \frac 32)_{(e-a,e-b,e-c)}}\cdot \frac {\Gamma(2)\Gamma(1)\Gamma({ \frac 32})}{\Gamma^3( \frac 32)}. \end{align}$

(3.1)

Substituting $\Gamma({ \frac 32}) = \frac {\sqrt{\pi}}2$ into (3.1) we obtain the result immediately. Putting $(a, b, c) = (0, 0, 0)$ in Theorem 3.1 we get the following general double summation formula for $1/\pi$ with two free parameters.

Corollary 3.2 If $d\in \Bbb N_0, Re(e-d+1) > 0$ and $Re(d+e+ \frac 32) > 0,$ then

$\sum\limits_{m,n = 0}^\infty \frac {( \frac 12)_{(m,d+n)}^3} {m!n!(d+1)_{m+n}(2)_{e+m}( \frac 32)_{d+e+n}} = \frac {4( \frac 12)_d^3(1)_{e-d}}{\pi( \frac 32)_e^3}.$

Setting $d = 0$ and $e = k\in\Bbb N_0$ in Corallary 3.2 we have the following result.

Proposition 3.3 Let $k$ be a nonnegative integer. Then

$\sum\limits_{m,n = 0}^\infty \frac {( \frac 12)_{(m,n)}^3}{m!n!(m+n)!(m+k+1)!( \frac 32+k)_n} = \frac {4k!}{\pi( \frac 32)_k^2}.$

Example 3.1 ( $k = 0$ in Proposition 3.3).

$\sum\limits_{m,n = 0}^\infty \frac {( \frac 12)_m^3( \frac 12)_n^2}{m!n!(m+n)!(m+1)!(2n+1)} = \frac 4{\pi}.$

If $d = e = k\in\Bbb N_0$ in Corollary 3.2 we achieve

Proposition 3.4 Let $k$ be a nonnegative integer. Then

$\sum\limits_{m,n = 0}^\infty \frac {( \frac 12)_{(m,n+k)}^3} {m!n!(k+1)_{m+n}(m+k+1)!( \frac 32)_{n+2k}} = \frac 4{\pi(2k+1)^3}.$

If we put $k = 0$ into Proposition 3.4, then we can also get Example 3.1.

4. Double series expansions for $\pi$

In this section we will prove the following theorem, which allows us to derive infinitely double series expansions for $\pi$ .

Theorem 4.1 If $d\in \Bbb N_0, Re(e-d-{\sigma}+1) > 0$ and $Re(d+e-a-b-c+2) > 0,$ then

$\begin{align*} &\sum\limits_{m,n = 0}^\infty \frac {({\sigma}-1)_{(a+m,b+m,c+m)}({\sigma})_{(d-a+n,d-b+n,d-c+n)}} {m!n!(2{\sigma}+d-1)_{m+n}({\sigma})_{e+m}(2)_{d+e-a-b-c+n}}\\ & = \frac {({\sigma}-1)_{(a,b,c)}({\sigma})_{(d-a,d-b,d-c)}(1-{\sigma})_{e-d}}{(1)_{(e-a,e-b,e-c)}} \cdot \frac {\pi}{\sin{{\sigma}\pi}}. \end{align*}$

Proof. Let $({\alpha}, \beta, {\gamma}, {\delta}) = ({\sigma}-1, {\sigma}-1, {\sigma}-1, 2{\sigma}-1)$ in Theorem 1.3. We obtain that

$\begin{align*} &\sum\limits_{m,n = 0}^\infty \frac {({\sigma}-1)_{(a+m,b+m,c+m)}({\sigma})_{(d-a+n,d-b+n,d-c+n)}} {m!n!(2{\sigma}+d-1)_{m+n}({\sigma})_{e+m}(2)_{d+e-a-b-c+n}}\\ & = \frac {({\sigma}-1)_{(a,b,c)}({\sigma})_{(d-a,d-b,d-c)}(1-{\sigma})_{e-d}} {(1)_{(e-a,e-b,e-c)}} \cdot \frac {\Gamma({\sigma})\Gamma({1-{\sigma}})\Gamma({2})}{\Gamma^3(1)}. \end{align*}$

(4.1)

Combining $\Gamma({{\sigma}})\Gamma({1-{\sigma}}) = \frac \pi{\sin{\sigma}\pi}$ with (4.1) we get the desired result immediately. Putting $a = b = c = 0$ in Theorem 4.1 we obtain the following equation.

Corollary 4.2 If $d\in \Bbb N_0, Re(e-d-{\sigma}+1) > 0$ and $Re(d+e+2) > 0,$ then

$\sum\limits_{m,n = 0}^\infty \frac {({\sigma}-1)_m^3({\sigma})_{d+n}^3} {m!n!(2{\sigma}+d-1)_{m+n}({\sigma})_{e+m}(2)_{d+e+n}} = \frac {({\sigma})_d^3(1-{\sigma})_{e-d}}{(1)_e^3} \cdot \frac {\pi}{\sin{{\sigma}\pi}}.$

4.1. $\sigma=\frac{1}{2}$ in Corollary 4.2

Letting $\sigma=\frac{1}{2}$ in Corollary 4.2, we get the following proposition.

Proposition 4.3 If $d\in \Bbb N_0, Re(e-d+ \frac 12) > 0$ and $Re(d+e+2) > 0,$ then

$\sum\limits_{m,n = 0}^\infty \frac {(- \frac 12)_m^3( \frac 12)_{d+n}^3}{m!n!(d)_{m+n}( \frac 12)_{e+m} (2)_{d+e+n}} = \frac {( \frac 12)_d^3( \frac 12)_{e-d}}{(1)_{e}^3} \pi.$

When we set $d = 1$ and $e = k\in\Bbb N = \{1, 2, 3\ldots\}$ in Proposition 4.3 we obtain

Proposition 4.4 If $k$ is a positive integer, then

$\sum\limits_{m,n = 0}^\infty \frac {(- \frac 12)_m^3( \frac 32)_n^3}{m!n!(m+n)!(n+k+2)!( \frac 12)_{m+k}} = \frac {\pi( \frac 12)_{k-1}}{(k!)^3} .$

Example 4.1 ( $k = 1$ in Proposition 4.4).

$\sum\limits_{m,n = 0}^\infty \frac {(- \frac 12)_m^3( \frac 32)_n^3}{m!n!(m+n)!(n+3)!( \frac 12)_{m+1}} = \pi.$

4.2. $\sigma=\frac{1}{3}$ in Corollary 4.2

Putting $\sigma=\frac{1}{3}$ in Corollary 4.2, we get the following proposition.

Proposition 4.5 If $d\in \Bbb N_0, Re(e-d+ \frac 23) > 0$ and $Re(d+e+2) > 0,$ then

$\sum\limits_{m,n = 0}^\infty \frac {(- \frac 23)_m^3( \frac 13)_{d+n}^3}{m!n!(d- \frac 13)_{m+n}( \frac 13)_{e+m} (2)_{d+e+n}} = \frac {2\sqrt3\pi( \frac 13)_d^3( \frac 23)_{e-d}}{3(1)_{e}^3}.$

When we set $d = 0$ and $e = k\in\Bbb N_0$ in Proposition 4.5 we obtain

Proposition 4.6 If $k$ is a nonnegative integer, then

$\sum\limits_{m,n = 0}^\infty \frac {(- \frac 23)_m^3( \frac 13)_n^3}{m!n!(- \frac 13)_{m+n}( \frac 13)_{m+k}(n+k+1)!} = \frac {2\sqrt3\pi( \frac 23)_k}{3k!^3}.$

Example 4.2 ( $k = 0$ in Proposition 4.6).

$\sum\limits_{m,n = 0}^\infty \frac {(- \frac 23)_m^2( \frac 13)_n^3}{m!n!(n+1)!(2-3m)(- \frac 13)_{m+n}} = \frac {\sqrt3\pi}3.$

Setting $d = e = k\in\Bbb N_0$ in Proposition 4.5, we get

Proposition 4.7 If $k$ is a nonnegative integer, then

$\sum\limits_{m,n = 0}^\infty \frac {(- \frac 23)_m^3( \frac 13+k)_n^3}{m!n!(n+2k+1)!(k- \frac 13)_{m+n}( \frac 13)_{m+k} } = \frac {2\sqrt3\pi}{3k!^3}.$

Therefore, Example 4.2 can also be deduced by fixing $k = 0$ in the above equation.

Example 4.3 ( $k = 1$ in Proposition 4.7).

$\sum\limits_{m,n = 0}^\infty \frac {(- \frac 23)_m^3( \frac 43)_n^3}{m!n!(n+3)!( \frac 23)_{m+n} ( \frac 43)_m} = \frac {2\sqrt3\pi}9.$

5. Conclusions

Double series expansions for $1/\pi$ and $\pi$ with several free parameters are established and many interesting formulas are obtained. A point that should be stressed is that there is an important connection between the summation formulas for double hypergeometric functions and double series expansions for the powers of $\pi$ .

Acknowledgements

The author was partially supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (grant 19KJB110006).

Conflicts of interest

The author declares that there is no conflict of interest in this paper.

References

[1]	G. E. Andrews, R. Askey, R. Roy, Special functions, Cambridge University Press, Cambridge, 1999.
[2]	P. Appell, J. K. de Fériet, Fonctions Hypergéométriques et Hypergéométriques: Polynômes d'Hermite, Gauthier-Villars, 1926.
[3]	W. N. Bailey, Generalized hypergeometric series, Cambridge University Press, Cambridge, 1935.
[4]	N. D. Baruah, B. C. Berndt, Ramanujan's series for $1/\pi$ arising from his cubic and quartic theories of elliptic functions, J. Math. Anal. Appl., 341 (2008), 357–371. doi: 10.1016/j.jmaa.2007.10.011
[5]	J. M. Borwein, P. B. Borwein, Pi and the AGM, Wiley, New York, 1987.
[6]	H. H. Chan, S. H. Chan, Z. Liu, Domb's numbers and Ramanujan-Sato type series for $1/\pi,$ Adv. Math., 186 (2004), 396–410.
[7]	W. Chu, $\pi$ -formulas implied by Dougall's summation theorem for $_5F_4$ -series, Ramanujan J., 26 (2011), 251–255. doi: 10.1007/s11139-010-9274-x
[8]	K. S. Chua, Extremal modular lattices, Mckay Thompson series, quagratic iterations, and new series for $1/\pi,$ Exp. Math., 14 (2005), 343–357.
[9]	J. Guillera, WZ pairs and $q$ -analogues of Ramanujan series for $1/\pi$ , J. Differ. Equ. Appl., 24 (2018), 1871–1879. doi: 10.1080/10236198.2018.1551380
[10]	J. Guillera, Bilateral Ramanujan-like series for $1/\pi^k$ and their congruences, Int. J. Number Theory, 16 (2020), 1969–1988. doi: 10.1142/S1793042120501018
[11]	J. Guillera, W. Zudilin, Ramanujan-type formulae for $1/\pi$ : the art of translation, Ramanujan Math. Soc. Lect. Notes Ser., 20 (2013), 181–195.
[12]	V. J. W. Guo, $q$ -Analogues of three Ramanujan-type formulas for $1/\pi$ , Ramanujan J., 52 (2020), 123–132. doi: 10.1007/s11139-018-0096-6
[13]	V. J. W. Guo, J. C. Liu, $q$ -Analogues of two Ramanujan-type formulas for $1/\pi$ , J. Differ. Equ. Appl., 24 (2018), 1368–1373. doi: 10.1080/10236198.2018.1485669
[14]	V. J. W. Guo, W. Zudilin, Ramanujan-type formulae for $1/\pi$ : $q$ -analogues, Integral Transforms Spec. Funct., 29 (2018), 505–513. doi: 10.1080/10652469.2018.1454448
[15]	Z. G. Liu, Gauss summation and Ramanujan type series for $1/\pi$ and $1/{\pi}^2,$ Int. J. Number Theory, 8 (2012), 289–297.
[16]	Z. G. Liu, A summation formula and Ramanujan type series, J. Math. Anal. Appl., 389 (2012), 1059–1065. doi: 10.1016/j.jmaa.2011.12.048
[17]	S. N. Pitre, J. Van der Jeugt, Transformation and summation formulas for Kampé de Fériet series $F_{1:1}^{0:3}(1, 1)$ , J. Math. Anal. Appl., 202 (1996), 121–132. doi: 10.1006/jmaa.1996.0306
[18]	S. Ramanujan, Modular equations and approximations to $\pi$ , Quart. J. Math. Oxford Ser., 45 (1914), 350–372.
[19]	H. M. Srivastava, P. W. Karlsson, Multiple Gaussian hypergeometric series, Halsted/Ellis Horwood/Wiley, New York/Chichester/New York, 1985.
[20]	C. Wei, $q$ -Analogues of several $\pi$ -formulas, Proc. Amer. Math. Soc., 148 (2020), 2287–2296. doi: 10.1090/proc/14664
[21]	C. Wei, D. Gong, Extensions of Ramanujan's two formulas for $1/\pi,$ J. Number Theory, 133 (2013), 2206–2216.
[22]	C. Wei, D. Gong, J. Li, $\pi$ -formulas with free parameters, J. Math. Anal. Appl., 396 (2012), 880–887. doi: 10.1016/j.jmaa.2012.07.039
[23]	R. M. Young, An introduction to nonharmonic Fourier series, Academic Press, New York, 1980.

Reader Comments

Your name:*

Email:*
© 2021 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Mathematics

1.8 3.4

Metrics

Article views(2264) PDF downloads(135) Cited by(0)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

AIMS Mathematics

Double series expansions for $\pi$

Related Papers:

Abstract

1. Introduction

2. Proof of Theorem 1.3

3. Double series expansions for $1/\pi$

4. Double series expansions for $\pi$

4.1. $\sigma=\frac{1}{2}$ in Corollary 4.2

4.2. $\sigma=\frac{1}{3}$ in Corollary 4.2

5. Conclusions

Acknowledgements

Conflicts of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Catalog

AIMS Mathematics

Double series expansions for π \pi

Related Papers:

Abstract

1. Introduction

2. Proof of Theorem 1.3

3. Double series expansions for 1/π 1/\pi

4. Double series expansions for π \pi

4.1. σ=12 \sigma=\frac{1}{2} in Corollary 4.2

4.2. σ=13 \sigma=\frac{1}{3} in Corollary 4.2

5. Conclusions

Acknowledgements

Conflicts of interest

References

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog

Double series expansions for $\pi$

3. Double series expansions for $1/\pi$

4. Double series expansions for $\pi$

4.1. $\sigma=\frac{1}{2}$ in Corollary 4.2

4.2. $\sigma=\frac{1}{3}$ in Corollary 4.2