$\pi$-Formulae from dual series of the Dougall theorem
UDC 517.5 By means of the extended Gould–Hsu inverse series relations, we find that the dual relation of Dougall's summation theorem for the well-poised $_7F_6$-series can be used to construct numerous interesting Ramanujan-like infinite-series express...
Saved in:
| Date: | 2023 |
|---|---|
| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2023
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/6587 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512455600373760 |
|---|---|
| author | Chu, W. Chu, W. |
| author_facet | Chu, W. Chu, W. |
| author_sort | Chu, W. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-02-05T14:53:26Z |
| description | UDC 517.5
By means of the extended Gould–Hsu inverse series relations, we find that the dual relation of Dougall's summation theorem for the well-poised $_7F_6$-series can be used to construct numerous interesting Ramanujan-like infinite-series expressions  for $\pi^{\pm1}$ and $\pi^{\pm2},$ including an elegant formula  for $\pi^{-2}$ due to Guillera. |
| doi_str_mv | 10.37863/umzh.v74i12.6587 |
| first_indexed | 2026-03-24T03:29:04Z |
| format | Article |
| fulltext |
DOI: 10.37863/umzh.v74i12.6587
UDC 517.5
W. Chu1 (School Math. and Statistics, Zhoukou Normal Univ., Henan, China and Univ. Salento, Italy)
\bfitpi -FORMULAE FROM DUAL SERIES OF THE DOUGALL THEOREM
\bfitpi -ФОРМУЛИ З ДУАЛЬНИХ РЯДIВ ТЕОРЕМИ ДУГАЛЛА
By means of the extended Gould – Hsu inverse series relations, we find that the dual relation of Dougall’s summation theorem
for the well-poised 7F6 -series can be used to construct numerous interesting Ramanujan-like infinite-series expressions for
\pi \pm 1 and \pi \pm 2, including an elegant formula for \pi - 2 due to Guillera.
За допомогою узагальнених спiввiдношень Гулда та Хсу для оберненого ряду доведено, що дуальне спiввiдношення
теореми пiдсумовування Дугалла для добре збалансованого 7F6 -ряду можна використати для побудови багатьох
цiкавих виразiв для нескiнченного ряду, подiбних до виразiв, що були отриманi Рамануджаном для \pi \pm 1 i \pi \pm 2,
включаючи елегантну формулу Гiльєра для \pi - 2 .
1. Introduction and motivation. In 1973, Gould and Hsu [27] discovered a useful pair of inverse
series relations, which can equivalently be reproduced below. Let \{ ai, bi\} be any two complex
sequences such that the \varphi -polynomials defined by
\varphi (x; 0) \equiv 1 and \varphi (x;n) =
n - 1\prod
k=0
(ak + xbk) for n \in \BbbN
differ from zero for x, n \in \BbbN 0. Then there hold the inverse series relations
f(n) =
n\sum
k=0
( - 1)k
\biggl(
n
k
\biggr)
\varphi (k;n) g(k),
g(n) =
n\sum
k=0
( - 1)k
\biggl(
n
k
\biggr)
ak + kbk
\varphi (n; k + 1)
f(k).
This inverse pair has wide applications to terminating hypergeometric series identities [9 – 12, 15, 24].
The duplicate form with applications can be found in [17, 18, 20]. There exist also q-analogues due
to Carlitz [6] which has applications to q-series identities [13, 14, 16, 19, 25, 26].
The Gould – Hsu inversions have the following extended form (cf. [4, 9, 15]):
f(n) =
n\sum
k=0
( - 1)k
\biggl(
n
k
\biggr)
\varphi (\lambda + k;n)\varphi ( - k;n)
\lambda + 2k
(\lambda + n)k+1
g(k), (1a)
g(n) =
n\sum
k=0
( - 1)k
\biggl(
n
k
\biggr)
(ak + \lambda bk + kbk)(ak - kbk)
\varphi (\lambda + n; k + 1)\varphi ( - n; k + 1)
(\lambda + k)nf(k), (1b)
where the shifted factorials are defined by
(x)0 = 1 and (x)n =
\Gamma (x+ n)
\Gamma (x)
= x(x+ 1) . . . (x+ n - 1) for n \in \BbbN .
1 e-mail: chu.wenchang@unisalento.it.
c\bigcirc W. CHU, 2022
1686 ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
\pi -FORMULAE FROM DUAL SERIES OF THE DOUGALL THEOREM 1687
There exist numerous hypergeometric series identities (see, for example, [5], Chapter 8, and [7 –
12, 15, 23, 24]). One of well-known summation theorems originally due to Dougall [22] is about
the terminating well-poised 7F6-series. By examining its dual formulae through (1a), (1b), we find
that their limiting relations result unexpectedly in \pi -related infinite series expressions, including the
following elegant formula discovered by Guillera [28 – 30]:
32
\pi 2
=
\infty \sum
k=0
\left[ 1
2
,
1
2
,
1
2
,
1
4
,
3
4
1, 1, 1, 1, 1
\right]
k
3 + 34k + 120k2
16k
.
By means of the duplicate forms of (1a), (1b), we shall work out, in details, the dual formulae of
Dougall’s summation theorem in the next section. Then applications will be presented in Section 3,
where several \pi -related infinite series of Ramanujan-like [32] with the convergence rate “
1
16
” will
be illustrated as examples.
Recall that the \Gamma -function (see, for example, [31], \S 8) is defined by the beta integral
\Gamma (x) =
\infty \int
0
ux - 1e - udu for \Re (x) > 0,
which admits Euler’s reflection property
\Gamma (x)\Gamma (1 - x) =
\pi
\mathrm{s}\mathrm{i}\mathrm{n}\pi x
with \Gamma
\biggl(
1
2
\biggr)
=
\surd
\pi .
The asymptotic formula
\Gamma (x+ n) \approx nx(n - 1)! as n \rightarrow \infty (2)
will be useful in evaluating limits of \Gamma -function quotients.
For the sake of brevity, the product and quotient of shifted factorials will respectively be abbre-
viated to
[\alpha , \beta , . . . , \gamma ]n = (\alpha )n(\beta )n . . . (\gamma )n,\Biggl[
\alpha , \beta , . . . , \gamma
A,B, . . . , C
\Biggr]
n
=
(\alpha )n(\beta )n . . . (\gamma )n
(A)n(B)n . . . (C)n
.
The similar notation will be employed for the \Gamma -function quotient
\Gamma
\Biggl[
\alpha , \beta , . . . , \gamma
A,B, . . . , C
\Biggr]
=
\Gamma (\alpha )\Gamma (\beta ) . . .\Gamma (\gamma )
\Gamma (A)\Gamma (B) . . .\Gamma (C)
.
2. Main theorems from duplicate inversions. The fundamental identity discovered by Dou-
gall [22] (see also [3], \S 4.3) for very well-poised terminating 7F6-series can be stated as
\Omega n(a; b, c, d) :=
\Biggl[
1 + a, 1 + a - b - c, 1 + a - b - d, 1 + a - c - d
1 + a - b, 1 + a - c, 1 + a - d, 1 + a - b - c - d
\Biggr]
n
=
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1688 W. CHU
=
n\sum
k=0
a+ 2k
a
\Biggl[
a, b, c, d, e, - n
1, 1 + a - b, 1 + a - c, 1 + a - d, 1 + a - e, 1 + a+ n
\Biggr]
k
, (3)
where the series is 2-balanced because 1 + 2a+ n = b+ c+ d+ e.
For all n \in \BbbN 0, it is well-known that n =
\Bigl\lfloor n
2
\Bigr\rfloor
+
\biggl\lfloor
1 + n
2
\biggr\rfloor
, where \lfloor x\rfloor denotes the greatest
integer not exceeding x. Then it is not difficult to check that Dougall’s formula (3) is equivalent to
the following one:
\Omega n
\biggl(
a; b+
\Bigl\lfloor n
2
\Bigr\rfloor
, c, d+
\biggl\lfloor
1 + n
2
\biggr\rfloor \biggr)
=
\Biggl[
1 + a - c - d, b+ c - a
1 + a - d, b - a
\Biggr]
\lfloor n
2 \rfloor
\times
\times
\Biggl[
1 + a, b+ d - a
1 + a - c, b+ c+ d - a
\Biggr]
n
\Biggl[
1 + a - b - c, c+ d - a
1 + a - b, d - a
\Biggr]
\lfloor 1+n
2 \rfloor
with its parameters subject to 1 + 2a = b+ c+ d+ e . Reformulate the above equality as a bino-
mial sum
n\sum
k=0
( - 1)k
\biggl(
n
k
\biggr)
[b+ k, b - a - k]\lfloor n
2 \rfloor [d+ k, d - a - k]\lfloor 1+n
2 \rfloor \times
\times a+ 2k
(a+ n)k+1
\Biggl[
a, b, c, d, 1 + 2a - b - c - d
1 + a - b, 1 + a - c, 1 + a - d, b+ c+ d - a
\Biggr]
k
=
=
\Biggl[
b, 1 + a - c - d, b+ c - a
1 + a - d
\Biggr]
\lfloor n
2 \rfloor
\Biggl[
d, 1 + a - b - c, c+ d - a
1 + a - b
\Biggr]
\lfloor 1+n
2 \rfloor
\times
\times
\Biggl[
a, b+ d - a
1 + a - c, b+ c+ d - a
\Biggr]
n
.
This equality matches exactly to (1a) under the assignments \lambda \rightarrow a and
\varphi (x;n) = (b - a+ x)\lfloor n
2 \rfloor (d - a+ x)\lfloor 1+n
2 \rfloor
as well as
f(n) =
\Biggl[
1 + a - c - d, b, b+ c - a
1 + a - d
\Biggr]
\lfloor n
2 \rfloor
\Biggl[
a, b+ d - a
1 + a - c, b+ c+ d - a
\Biggr]
n
\times
\times
\Biggl[
1 + a - b - c, d, c+ d - a
1 + a - b
\Biggr]
\lfloor 1+n
2 \rfloor
,
g(k) =
\Biggl[
a, b, c, d, 1 + 2a - b - c - d
1 + a - b, 1 + a - c, 1 + a - d, b+ c+ d - a
\Biggr]
k
.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
\pi -FORMULAE FROM DUAL SERIES OF THE DOUGALL THEOREM 1689
The dual relation corresponding to (1b) can explicitly be stated, according to the parity of k and
(a)k(a+ k)n = (a)n(a+ n)k, as\Biggl[
b, c, d, 1 + 2a - b - c - d
1 + a - b, 1 + a - c, 1 + a - d, b+ c+ d - a
\Biggr]
n
=
=
\sum
k\geq 0
\biggl(
n
2k
\biggr)
(d+ 3k)(d - a - k)(a+ n)2k
[b+ n, b - a - n]k[d+ n, d - a - n]k+1
\times
\times
\Biggl[
1 + a - c - d, b, b+ c - a
1 + a - d
\Biggr]
k
\Biggl[
1 + a - b - c, d, c+ d - a
1 + a - b
\Biggr]
k
\times
\times
\Biggl[
b+ d - a
1 + a - c, b+ c+ d - a
\Biggr]
2k
-
-
\sum
k\geq 0
\biggl(
n
2k + 1
\biggr)
(b+ 3k + 1)(b - a - k - 1)(a+ n)2k+1
[b+ n, b - a - n]k+1[d+ n, d - a - n]k+1
\times
\times
\Biggl[
1 + a - c - d, b, b+ c - a
1 + a - d
\Biggr]
k
\Biggl[
1 + a - b - c, d, c+ d - a
1 + a - b
\Biggr]
k+1
\times
\times
\Biggl[
b+ d - a
1 + a - c, b+ c+ d - a
\Biggr]
2k+1
.
Now multiplying by “n2” across this binomial relation and then letting n \rightarrow \infty , we may evaluate
the limits of the left member by (2) and of the corresponding right member through the Weierstrass
M -test on uniformly convergent series (cf. [33], \S 3.106). After some routine simplification, the
resulting limiting relation can be expressed explicitly in the following lemma.
Lemma 1 (infinite series identity).
\Gamma
\Biggl[
1 + a - b, 1 + a - c, 1 + a - d, b+ c+ d - a
b, c, d, 1 + 2a - b - c - d
\Biggr]
=
=
\sum
k\geq 0
(d+ 3k)(a - d)
(2k)!
\Biggl[
b+ d - a
1 + a - c, b+ c+ d - a
\Biggr]
2k
\times
\times
\Biggl[
1 + a - c - d, b, b+ c - a
a - d
\Biggr]
k
\Biggl[
1 + a - b - c, d, c+ d - a
1 + a - b
\Biggr]
k
+
+
\sum
k\geq 0
(b+ 3k + 1)(a - b)
(2k + 1)!
\Biggl[
b+ d - a
1 + a - c, b+ c+ d - a
\Biggr]
2k+1
\times
\times
\Biggl[
1 + a - c - d, b, b+ c - a
1 + a - d
\Biggr]
k
\Biggl[
1 + a - b - c, d, c+ d - a
a - b
\Biggr]
k+1
.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1690 W. CHU
According to this lemma, we are going to show two main theorems that will be utilized, in the
next section, to deduce infinite series expressions for \pi \pm 1 and \pi \pm 2.
For the equality in Lemma 1, multiplying both sides by (1 + a - c)(b + c + d - a) and then
unifying the two sums, we derive the following infinite series identity.
Theorem 1 (infinite series identity).
\Gamma
\Biggl[
1 + a - b, 2 + a - c, 1 + a - d, 1 - a+ b+ c+ d
b, c, d, 1 + 2a - b - c - d
\Biggr]
=
=
\infty \sum
k=0
\scrP (k)
[b, d, 1 + a - b - c, 1 + a - c - d, b+ c - a, c+ d - a]k(b+ d - a)2k
(2k + 1)! [1 + a - b, 1 + a - d]k[2 + a - c, 1 - a+ b+ c+ d]2k
,
where \scrP (k) is the polynomial given by
\scrP (k) = (1 + a - b - c+ k)(d+ k)(c+ d - a+ k)(b+ d - a+ 2k)(1 + b+ 3k)+
+(1 + 2k)(a - d+ k)(1 + a - c+ 2k)(b+ c+ d - a+ 2k)(d+ 3k).
Alternatively, by shifting backward k \rightarrow k - 1 for the second sum and then unifying it to the
first one, we get analogously, from Lemma 2 another infinite series identity.
Theorem 2 (infinite series identity).
\Gamma
\Biggl[
1 + a - b, 1 + a - c, 1 + a - d, b+ c+ d - a
b, c, d, 1 + 2a - b - c - d
\Biggr]
=
=
\infty \sum
k=0
\scrQ (k)
[b, d, 1 + a - b - c, 1 + a - c - d, b+ c - a, c+ d - a]k(b+ d - a)2k
(2k)! [1 + a - b, 1 + a - d]k[1 + a - c, b+ c+ d - a]2k
,
where \scrQ (k) is the rational function given by
\scrQ (k) = (a - d+ k)(d+ 3k)\times
\times
\biggl\{
1 +
(2k)(a - b+ k)(a - c+ 2k)(b+ c+ d - a - 1 + 2k)(b - 2 + 3k)
(a - c - d+ k)(b - 1 + k)(b+ c - a - 1 + k)(b+ d - a - 1 + 2k)(d+ 3k)
\biggr\}
.
3. Infinite series for \bfitpi \pm \bfone and \bfitpi \pm \bftwo . By applying Theorems 1 and 2, we can derive numerous
infinite series identities. They are recorded below in seven classes whose weight polynomial degrees
are not greater than 3. For all the examples, the parameter settings a, b, c, d and eventual references
are highlight in their headers. In order to ensure the accuracy, all the summation formulae in this
section are verified experimentally by appropriately devised Mathematica commands.
3.1. Series for \bfitpi - \bftwo .
Example 1
\Biggl(
Guillera [28 – 30]:
1
2
,
1
2
,
1
2
,
1
2
in Theorem 1
\Biggr)
:
32
\pi 2
=
\infty \sum
k=0
\left[ 1
2
,
1
2
,
1
2
,
1
4
,
3
4
1, 1, 1, 1, 1
\right]
k
120k2 + 34k + 3
16k
.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
\pi -FORMULAE FROM DUAL SERIES OF THE DOUGALL THEOREM 1691
Example 2
\Biggl(
Chu and Zhang [21]:
3
2
,
1
2
,
3
2
,
1
2
in Theorem 1
\Biggr)
:
128
\pi 2
=
\infty \sum
k=0
\left[ 1
2
,
1
2
,
1
2
,
1
4
, - 1
4
1, 1, 1, 2, 2
\right]
k
120k2 + 118k + 13
16k
.
Example 3
\Biggl(
3
2
,
1
2
,
1
2
,
3
2
in Theorem 1
\Biggr)
:
256
3\pi 2
=
\infty \sum
k=0
\left[ 1
2
, - 1
2
,
3
2
,
1
4
,
3
4
1, 1, 1, 2, 2
\right]
k
80k3 + 148k2 + 80k + 9
16k
.
Example 4
\Biggl(
3
2
,
3
2
,
1
2
,
3
2
in Theorem 1
\Biggr)
:
512
\pi 2
=
\infty \sum
k=0
\left[ 1
2
,
1
2
,
3
2
,
3
4
,
5
4
1, 1, 1, 2, 2
\right]
k
240k3 + 532k2 + 336k + 45
16k
.
Example 5
\Biggl(
1
2
,
1
2
,
1
2
, - 1
2
in Theorem 1
\Biggr)
:
32
\pi 2
=
\infty \sum
k=0
\left[
3
2
, - 1
2
, - 1
2
,
1
4
, - 1
4
,
7
6
1, 1, 1, 1, 2,
1
6
\right]
k
3 - 10k - 40k2
16k
.
Example 6
\Biggl(
3
2
,
1
2
,
3
2
, - 1
2
in Theorem 1
\Biggr)
:
256
3\pi 2
=
\infty \sum
k=0
\left[
3
2
, - 1
2
, - 1
2
, - 1
4
, - 3
4
,
7
6
1, 1, 1, 2, 3,
1
6
\right]
k
9 - 38k - 40k2
16k
.
Example 7
\Biggl(
3
2
,
3
2
,
1
2
,
3
2
in Theorem 2
\Biggr)
:
8
\pi 2
=
\infty \sum
k=1
\left[
3
2
, - 1
2
, - 1
2
,
1
4
,
3
4
,
7
6
1, 1, 1, 1, 1,
1
6
\right]
k
k(3 - 18k + 40k2)
16k
.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1692 W. CHU
Example 8
\Biggl(
3
2
,
1
2
,
3
2
,
1
2
in Theorem 2
\Biggr)
:
24
\pi 2
=
\infty \sum
k=0
\left[
1
2
, - 1
2
, - 1
2
, - 1
4
, - 3
4
,
5
4
,
5
6
1, 1, 1, 1, 2,
1
4
, - 1
6
\right]
k
3 + 8k + 20k2
16k
.
Example 9
\Biggl(
3
2
,
3
2
, - 1
2
,
1
2
in Theorem 1
\Biggr)
:
256
9\pi 2
=
\infty \sum
k=0
\left[
3
2
,
5
2
, - 1
2
, - 3
2
,
1
4
,
3
4
1, 1, 1, 2, 2,
1
2
\right]
k
5 + 12k - 68k2 - 80k3
16k
.
3.2. Series for \bfitpi \bftwo .
Example 10
\Biggl(
Chu and Zhang [21]:
3
2
, 1, 1, 1 in Theorem 1
\Biggr)
:
9\pi 2
8
=
\infty \sum
k=0
\left[ 1,
1
2
,
1
2
,
1
4
,
3
4
3
2
,
5
4
,
5
4
,
7
4
,
7
4
\right]
k
11 + 64k + 111k2 + 60k3
16k
.
Example 11
\Biggl(
5
2
, 2, 1, 2 in Theorem 1
\Biggr)
:
225\pi 2
32
=
\infty \sum
k=0
\left[ 2, 2,
1
2
,
1
2
,
3
4
,
5
4
1,
3
2
,
7
4
,
7
4
,
9
4
,
9
4
\right]
k
68 + 206k + 197k2 + 60k3
16k
.
Example 12
\Biggl(
5
2
, 1, 2, 2 in Theorem 1
\Biggr)
:
135\pi 2
64
=
\infty \sum
k=0
\left[ 2,
1
2
, - 1
2
,
5
3
,
1
4
,
3
4
5
2
,
2
3
,
5
4
,
7
4
,
7
4
,
9
4
\right]
k
21 + 93k + 110k2 + 40k3
16k
.
Example 13
\Biggl(
5
2
, 1, 2, 1 in Theorem 2
\Biggr)
:
3\pi 2
32
= 1 +
\infty \sum
k=1
\left[ 1, - 1
2
, - 1
2
, - 1
4
, - 3
4
5
2
,
3
4
,
3
4
,
5
4
,
5
4
\right]
k
3 + 3k - 22k2 - 40k3
16k
.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
\pi -FORMULAE FROM DUAL SERIES OF THE DOUGALL THEOREM 1693
Example 14
\Biggl(
7
2
, 1, 2, 1 in Theorem 2
\Biggr)
:
15\pi 2
256
=
1
3
+
\infty \sum
k=1
\left[ 1, - 1
2
, - 3
2
, - 3
4
, - 5
4
7
2
,
1
4
,
3
4
,
5
4
,
7
4
\right]
k
1 - 3k + 2k2 + 8k3
16k
.
Example 15
\Biggl(
7
2
, 2, 2, 2 in Theorem 2
\Biggr)
:
27\pi 2
128
=
\infty \sum
k=0
\left[ 2, - 1
2
, - 1
2
,
4
3
,
1
4
, - 1
4
5
2
,
1
3
,
5
4
,
5
4
,
7
4
,
7
4
\right]
k
2 - 21k - 66k2 - 40k3
16k
.
Example 16
\Biggl(
7
2
, 1, 2, 2 in Theorem 2
\Biggr)
:
405\pi 2
256
= 18 +
\infty \sum
k=1
\left[ 2, - 1
2
,
3
2
, - 3
2
, - 1
4
, - 3
4
1
2
,
7
2
,
3
4
,
5
4
,
5
4
,
7
4
\right]
k
48 - 59k - 194k2 - 120k3
16k
.
3.3. Series for \bfitpi \bftwo /\bfGamma \bfthree .
Example 17
\Biggl(
1
2
,
1
3
,
1
3
, - 2
3
in Theorem 1
\Biggr)
:
98\pi 2
3\Gamma
\biggl(
2
3
\biggr) 3 =
\infty \sum
k=0
\left[
1
3
, - 2
3
,
5
6
, - 5
6
,
11
6
,
10
9
,
1
12
, - 5
12
1,
3
2
,
1
4
,
3
4
,
13
6
,
1
9
,
13
12
,
19
12
\right]
k
118 + 45k - 1098k2 - 1080k3
16k
.
Example 18
\Biggl(
3
2
,
1
3
,
1
3
,
4
3
in Theorem 1
\Biggr)
:
637\pi 2
16\Gamma
\biggl(
2
3
\biggr) 3 =
\infty \sum
k=0
\left[
1
3
,
4
3
,
5
6
, - 5
6
,
11
6
,
13
9
,
1
12
,
7
12
1,
3
2
,
3
4
,
5
4
,
13
6
,
4
9
,
19
12
,
25
12
\right]
k
1080k3 + 2286k2 + 1395k + 161
16k
.
Example 19
\Biggl(
1
2
,
2
3
, - 1
3
, - 1
3
in Theorem 1
\Biggr)
:
275\pi 2
\Gamma
\biggl(
1
3
\biggr) 3 =
\infty \sum
k=0
\left[
2
3
, - 1
3
,
7
6
, - 7
6
,
13
6
,
11
9
, - 1
12
,
5
12
1,
3
2
,
1
4
,
3
4
,
11
6
,
2
9
,
17
12
,
23
12
\right]
k
125 - 351k - 1602k2 - 1080k3
16k
.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1694 W. CHU
Example 20
\Biggl(
3
2
,
2
3
,
5
3
, - 1
3
in Theorem 1
\Biggr)
:
825\pi 2
8\Gamma
\biggl(
1
3
\biggr) 3 =
\infty \sum
k=0
\left[
2
3
, - 1
3
,
1
6
, - 1
6
,
7
6
,
11
9
, - 1
12
, - 7
12
1,
3
2
,
3
4
,
5
4
,
17
6
,
2
9
,
11
12
,
17
12
\right]
k
53 - 315k - 1278k2 - 1080k3
16k
.
Example 21
\Biggl(
3
2
,
2
3
, - 1
3
,
5
3
in Theorem 1
\Biggr)
:
2805\pi 2
4\Gamma
\biggl(
1
3
\biggr) 3 =
\infty \sum
k=0
\left[
2
3
,
5
3
,
7
6
, - 7
6
,
13
6
,
14
9
,
5
12
,
11
12
1,
3
2
,
3
4
,
5
4
,
11
6
,
5
9
,
23
12
,
29
12
\right]
k
1080k3 + 2790k2 + 2151k + 478
16k
.
Example 22
\Biggl(
- 1
2
, - 2
3
, - 2
3
, - 2
3
in Theorem 2
\Biggr)
:
3872\pi 2
243\Gamma
\biggl(
2
3
\biggr) 3 =
\infty \sum
k=0
\left[ - 2
3
, - 5
3
,
5
6
,
11
6
, - 11
6
,
4
9
, - 5
12
, - 11
12
1,
1
2
, - 1
4
, - 3
4
,
7
6
, - 5
9
,
7
12
,
13
12
\right]
k
1080k3 - 954k2 - 585k + 242
16k
.
Example 23
\Biggl(
1
2
, - 2
3
,
4
3
, - 2
3
in Theorem 2
\Biggr)
:
2380\pi 2
27\Gamma
\biggl(
2
3
\biggr) 3 =
\infty \sum
k=0
\left[ - 2
3
, - 5
3
, - 1
6
, - 5
6
,
5
6
,
4
9
, - 11
12
, - 17
12
1,
1
2
,
1
4
, - 1
4
,
13
6
, - 5
9
,
1
12
,
7
12
\right]
k
1080k3 - 1278k2 + 99k + 170
16k
.
Example 24
\Biggl(
1
2
,
4
3
, - 2
3
, - 2
3
in Theorem 2
\Biggr)
:
770\pi 2
27\Gamma
\biggl(
2
3
\biggr) 3 =
\infty \sum
k=0
\left[
1
3
, - 2
3
,
5
6
,
11
6
, - 11
6
,
7
9
, - 5
12
,
1
12
1,
1
2
,
1
4
, - 1
4
,
7
6
, - 2
9
,
13
12
,
19
12
\right]
k
55 + 441k - 234k2 - 1080k3
16k
.
Example 25
\Biggl(
3
2
,
4
3
, - 2
3
,
4
3
in Theorem 2
\Biggr)
:
- 1001\pi 2
18\Gamma
\biggl(
2
3
\biggr) 3 =
\infty \sum
k=0
\left[
1
3
,
4
3
,
5
6
,
11
6
, - 11
6
,
10
9
,
1
12
,
7
12
1,
1
2
,
1
4
,
3
4
,
7
6
,
1
9
,
19
12
,
25
12
\right]
k
1080k3 + 1422k2 + 351k + 44
16k
.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
\pi -FORMULAE FROM DUAL SERIES OF THE DOUGALL THEOREM 1695
Example 26
\Biggl(
3
2
,
4
3
,
4
3
, - 2
3
in Theorem 2
\Biggr)
:
385\pi 2
36\Gamma
\biggl(
2
3
\biggr) 3 =
\infty \sum
k=0
\left[
1
3
, - 2
3
, - 1
6
, - 5
6
,
5
6
,
7
9
, - 5
12
, - 11
12
1,
1
2
,
1
4
,
3
4
,
13
6
, - 2
9
,
7
12
,
13
12
\right]
k
55 + 189k + 90k2 - 1080k3
16k
.
Example 27
\Biggl(
1
2
, - 1
3
,
2
3
, - 1
3
in Theorem 2
\Biggr)
:
910\pi 2
9\Gamma
\biggl(
1
3
\biggr) 3 =
\infty \sum
k=0
\left[ - 1
3
, - 4
3
,
1
6
,
7
6
, - 7
6
,
5
9
, - 7
12
, - 13
12
1,
1
2
,
1
4
, - 1
4
,
11
6
, - 4
9
,
5
12
,
11
12
\right]
k
1080k3 - 774k2 - 225k + 91
16k
.
Example 28
\Biggl(
3
2
,
2
3
,
2
3
,
2
3
in Theorem 2
\Biggr)
:
1225\pi 2
6\Gamma
\biggl(
1
3
\biggr) 3 =
\infty \sum
k=0
\left[
2
3
, - 1
3
,
1
6
,
7
6
, - 7
6
,
8
9
, - 1
12
, - 7
12
1,
1
2
,
1
4
,
3
4
,
11
6
, - 1
9
,
11
12
,
17
12
\right]
k
98 + 153k - 414k2 - 1080k3
16k
.
3.4. Series for \bfGamma \bfthree /\bfitpi \bftwo .
Example 29
\Biggl(
- 1
2
, - 5
6
,
1
6
,
1
6
in Theorem 1
\Biggr)
:
180\Gamma
\biggl(
2
3
\biggr) 3
\pi 2
=
\infty \sum
k=0
\left[
1
6
, - 1
6
,
5
6
, - 5
6
,
7
6
, - 1
12
,
5
12
,
19
18
1, 1,
1
2
,
3
2
,
1
3
,
2
3
,
4
3
,
1
18
\right]
k
35 + 228k - 540k2 - 2160k3
16k
.
Example 30
\Biggl(
- 1
2
,
1
6
,
1
6
,
7
6
in Theorem 1
\Biggr)
:
8748\Gamma
\biggl(
2
3
\biggr) 3
7\pi 2
=
\infty \sum
k=0
\left[
1
6
,
5
6
, - 5
6
,
7
6
,
11
6
,
11
12
,
17
12
,
25
18
1, 2,
3
2
,
3
2
,
1
3
,
2
3
, - 2
3
,
7
18
\right]
k
593 + 1344k - 1404k2 - 2160k3
16k
.
Example 31
\Biggl(
1
2
,
1
6
, - 5
6
,
7
6
in Theorem 1
\Biggr)
:
960\Gamma
\biggl(
2
3
\biggr) 3
7\pi 2
=
\infty \sum
k=0
\left[
1
6
, - 1
6
,
7
6
, - 7
6
,
13
6
,
5
12
,
11
12
,
25
18
1, 1,
1
2
,
3
2
,
1
3
,
4
3
,
5
3
,
7
18
\right]
k
65 + 372k - 756k2 - 2160k3
16k
.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1696 W. CHU
Example 32
\Biggl(
1
2
,
1
6
,
7
6
, - 5
6
in Theorem 1
\Biggr)
:
960\Gamma
\biggl(
2
3
\biggr) 3
\pi 2
=
\infty \sum
k=0
\left[
1
6
, - 1
6
,
5
6
, - 5
6
,
7
6
, - 1
12
, - 7
12
,
19
18
1, 1,
1
2
,
3
2
,
2
3
,
4
3
,
7
3
,
1
18
\right]
k
245 - 204k - 2052k2 - 2160k3
16k
.
Example 33
\Biggl(
1
2
,
1
6
,
7
6
,
7
6
in Theorem 1
\Biggr)
:
7776\Gamma
\biggl(
2
3
\biggr) 3
7\pi 2
=
\infty \sum
k=0
\left[
1
6
,
5
6
,
5
6
, - 5
6
,
7
6
,
11
6
,
5
12
,
11
12
,
25
18
1, 2,
3
2
,
3
2
,
1
3
,
2
3
,
4
3
, - 1
6
,
7
18
\right]
k
360k2 + 546k + 191
16k
.
Example 34
\Biggl(
3
2
,
1
6
,
1
6
,
7
6
in Theorem 1
\Biggr)
:
1024\Gamma
\biggl(
2
3
\biggr) 3
21\pi 2
=
\infty \sum
k=0
\left[
1
6
, - 1
6
,
7
6
, - 7
6
,
13
6
, - 1
12
,
5
12
,
25
18
1, 1,
1
2
,
3
2
,
4
3
,
5
3
,
7
3
,
7
18
\right]
k
2160k3 + 2268k2 + 60k + 13
16k
.
Example 35
\Biggl(
- 1
2
, - 1
6
,
5
6
,
5
6
in Theorem 1
\Biggr)
:
2916\Gamma
\biggl(
1
3
\biggr) 3
5\pi 2
=
\infty \sum
k=0
\left[ - 1
6
,
5
6
,
7
6
, - 7
6
,
13
6
,
7
12
,
13
12
,
23
18
1, 2,
3
2
,
3
2
, - 1
3
,
1
3
,
2
3
,
5
18
\right]
k
697 + 1056k - 1836k2 - 2160k3
16k
.
Example 36
\Biggl(
1
2
, - 1
6
,
11
6
,
5
6
in Theorem 1
\Biggr)
:
2592\Gamma
\biggl(
1
3
\biggr) 3
25\pi 2
=
\infty \sum
k=0
\left[ - 1
6
,
5
6
,
7
6
, - 7
6
,
13
6
,
1
12
,
7
12
,
23
18
1, 2,
3
2
,
3
2
,
1
3
,
2
3
,
5
3
,
5
18
\right]
k
223 - 888k - 3348k2 - 2160k3
16k
.
Example 37
\Biggl(
1
2
,
5
6
, - 1
6
, - 1
6
in Theorem 1
\Biggr)
:
32\Gamma
\biggl(
1
3
\biggr) 3
5\pi 2
=
\infty \sum
k=0
\left[
1
6
, - 1
6
,
5
6
, - 5
6
,
11
6
,
1
12
,
7
12
,
23
18
1, 1,
1
2
,
3
2
,
2
3
,
4
3
,
5
3
,
5
18
\right]
k
2160k3 + 1188k2 - 84k + 11
16k
.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
\pi -FORMULAE FROM DUAL SERIES OF THE DOUGALL THEOREM 1697
Example 38
\Biggl(
1
2
,
5
6
, - 1
6
,
11
6
in Theorem 1
\Biggr)
:
2592\Gamma
\biggl(
1
3
\biggr) 3
55\pi 2
=
\infty \sum
k=0
\left[
1
6
, - 1
6
,
5
6
,
7
6
,
11
6
,
13
12
,
19
12
,
29
18
1, 2,
3
2
,
3
2
,
2
3
,
4
3
, - 1
3
,
11
18
\right]
k
151 + 264k - 2052k2 - 2160k3
16k
.
Example 39
\Biggl(
3
2
, - 1
6
,
5
6
,
5
6
in Theorem 1
\Biggr)
:
256\Gamma
\biggl(
1
3
\biggr) 3
9\pi 2
=
\infty \sum
k=0
\left[
1
6
, - 1
6
,
5
6
, - 5
6
,
11
6
,
1
12
, - 5
12
,
23
18
1, 1,
1
2
,
3
2
,
4
3
,
5
3
,
8
3
,
5
18
\right]
k
55 - 348k - 2700k2 - 2160k3
16k
.
Example 40
\Biggl(
3
2
,
5
6
,
5
6
,
11
6
in Theorem 1
\Biggr)
:
6912\Gamma
\biggl(
1
3
\biggr) 3
55\pi 2
=
\infty \sum
k=0
\left[
1
6
, - 1
6
,
5
6
,
7
6
,
11
6
,
7
12
,
13
12
,
29
18
1, 2,
3
2
,
3
2
,
2
3
,
4
3
,
5
3
,
11
18
\right]
k
2160k3 + 3564k2 + 1680k + 251
16k
.
Example 41
\Biggl(
- 1
2
,
1
6
,
7
6
,
1
6
in Theorem 2
\Biggr)
:
2673\Gamma
\biggl(
2
3
\biggr) 3
16\pi 2
=
\infty \sum
k=0
\left[
1
6
,
5
6
, - 5
6
,
11
6
, - 11
6
, - 1
12
,
5
12
,
13
18
1, 1,
1
2
,
3
2
,
1
3
, - 1
3
, - 2
3
, - 5
18
\right]
k
11 + 1380k + 1188k2 - 2160k3
16k
.
Example 42
\Biggl(
- 1
2
,
7
6
, - 5
6
,
7
6
in Theorem 2
\Biggr)
:
13365\Gamma
\biggl(
2
3
\biggr) 3
16\pi 2
=
\infty \sum
k=0
\left[
1
6
, - 1
6
,
5
6
, - 5
6
,
7
6
,
11
12
,
17
12
,
19
18
1, 1,
1
2
,
3
2
,
2
3
, - 2
3
, - 5
3
,
1
18
\right]
k
2160k3 - 2484k2 - 1092k + 385
16k
.
Example 43
\Biggl(
1
2
,
7
6
,
1
6
,
7
6
in Theorem 2
\Biggr)
:
675\Gamma
\biggl(
2
3
\biggr) 3
2\pi 2
=
\infty \sum
k=0
\left[
1
6
, - 1
6
,
5
6
, - 5
6
,
7
6
,
5
12
,
11
12
,
19
18
1, 1,
1
2
,
3
2
,
1
3
,
2
3
, - 2
3
,
1
18
\right]
k
2160k3 - 972k2 - 660k + 175
16k
.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1698 W. CHU
Example 44
\Biggl(
3
2
,
7
6
,
7
6
,
7
6
in Theorem 2
\Biggr)
:
180\Gamma
\biggl(
2
3
\biggr) 3
\pi 2
=
\infty \sum
k=0
\left[
1
6
, - 1
6
,
5
6
, - 5
6
,
7
6
, - 1
12
,
5
12
,
19
18
1, 1,
1
2
,
3
2
,
1
3
,
2
3
,
4
3
,
1
18
\right]
k
35 + 228k - 540k2 - 2160k3
16k
.
Example 45
\Biggl(
- 1
2
, - 1
6
,
11
6
, - 1
6
in Theorem 2
\Biggr)
:
1053\Gamma
\biggl(
1
3
\biggr) 3
32\pi 2
=
\infty \sum
k=0
\left[ - 1
6
,
7
6
, - 7
6
,
13
6
, - 13
6
, - 5
12
,
1
12
,
11
18
1, 1,
1
2
,
3
2
, - 1
3
, - 2
3
,
2
3
, - 7
18
\right]
k
2160k3 - 756k2 - 1668k + 65
16k
.
Example 46
\Biggl(
- 1
2
,
5
6
, - 1
6
,
5
6
in Theorem 2
\Biggr)
:
3969\Gamma
\biggl(
1
3
\biggr) 3
32\pi 2
=
\infty \sum
k=0
\left[
1
6
, - 1
6
,
5
6
,
7
6
, - 7
6
,
7
12
,
13
12
,
17
18
1, 1,
1
2
,
3
2
,
1
3
, - 1
3
, - 4
3
, - 1
18
\right]
k
2160k3 - 2052k2 - 1092k + 245
16k
.
Example 47
\Biggl(
1
2
,
5
6
,
5
6
,
5
6
in Theorem 2
\Biggr)
:
63\Gamma
\biggl(
1
3
\biggr) 3
10\pi 2
=
\infty \sum
k=0
\left[
1
6
, - 1
6
,
5
6
,
7
6
, - 7
6
,
1
12
,
7
12
,
17
18
1, 1,
1
2
,
3
2
, - 1
3
,
1
3
,
2
3
, - 1
18
\right]
k
432k3 - 108k2 - 132k + 7
16k
.
Example 48
\Biggl(
3
2
,
11
6
, - 1
6
,
11
6
in Theorem 2
\Biggr)
:
84\Gamma
\biggl(
1
3
\biggr) 3
5\pi 2
=
\infty \sum
k=0
\left[
1
6
, - 1
6
,
5
6
, - 5
6
,
11
6
,
7
12
,
13
12
,
23
18
1, 1,
1
2
,
3
2
, - 1
3
,
2
3
,
4
3
,
5
18
\right]
k
2160k3 - 324k2 - 516k + 77
16k
.
Example 49
\Biggl(
3
2
,
11
6
,
11
6
, - 1
6
in Theorem 2
\Biggr)
:
84\Gamma
\biggl(
1
3
\biggr) 3
\pi 2
=
\infty \sum
k=0
\left[
1
6
, - 1
6
,
5
6
,
7
6
, - 7
6
,
1
12
, - 5
12
,
17
18
1, 1,
1
2
,
3
2
,
1
3
,
2
3
,
5
3
, - 1
18
\right]
k
175 + 228k - 972k2 - 2160k3
16k
.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
\pi -FORMULAE FROM DUAL SERIES OF THE DOUGALL THEOREM 1699
3.5. Series for \bfitpi - \bfone .
Example 50
\Biggl(
Chu and Zhang [21]:
1
2
,
1
2
,
1
2
,
1
3
in Theorem 1
\Biggr)
:
15
\surd
3
\pi
=
\infty \sum
k=0
\left[
1
2
,
1
3
,
1
3
,
2
3
,
2
3
1, 1, 1,
11
12
,
17
12
\right]
k
135k2 + 75k + 8
16k
.
Example 51
\Biggl(
1
2
,
1
2
,
1
2
,
2
3
in Theorem 1
\Biggr)
:
21
\surd
3
\pi
=
\infty \sum
k=0
\left[
1
2
,
1
3
,
1
3
,
2
3
,
2
3
1, 1, 1,
13
12
,
19
12
\right]
k
810k3 + 684k2 + 141k + 10
16k
.
Example 52
\Biggl(
1
2
,
1
2
,
1
2
,
1
4
in Theorem 1
\Biggr)
:
48
\pi
=
\infty \sum
k=0
\left[
1
2
,
1
4
,
3
4
,
1
8
,
5
8
1, 1, 1,
7
8
,
11
8
\right]
k
480k2 + 212k + 15
16k
.
Example 53
\Biggl(
1
2
,
1
2
,
1
2
,
3
4
in Theorem 1
\Biggr)
:
80
3\pi
=
\infty \sum
k=0
\left[
1
2
,
1
4
,
3
4
,
3
8
,
7
8
1, 1, 1,
9
8
,
13
8
\right]
k
640k3 + 560k2 + 112k + 7
16k
.
Example 54
\Biggl(
1
2
,
1
2
,
1
6
,
1
2
in Theorem 1
\Biggr)
:
256
3\pi
\surd
3
=
\infty \sum
k=0
\left[
1
2
,
1
4
,
3
4
,
1
6
,
5
6
1, 1, 1,
4
3
,
5
3
\right]
k
720k3 + 804k2 + 236k + 15
16k
.
Example 55
\Biggl(
1
2
,
1
2
,
1
2
,
1
6
in Theorem 1
\Biggr)
:
192
\pi
\surd
3
=
\infty \sum
k=0
\left[
1
2
,
1
6
,
1
6
,
1
12
,
7
12
1, 1, 1,
4
3
,
4
3
\right]
k
6480k3 + 4284k2 + 840k + 35
16k
.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1700 W. CHU
Example 56
\Biggl(
Chu and Zhang [21]:
1
2
,
1
2
,
1
2
,
5
6
in Theorem 1
\Biggr)
:
384
\pi
\surd
3
=
\infty \sum
k=0
\left[
1
2
,
5
6
,
5
6
,
5
12
,
11
12
1, 1, 1,
2
3
,
5
3
\right]
k
1080k2 + 798k + 55
16k
.
Example 57
\Biggl(
1
2
,
1
2
,
1
2
,
1
5
in Theorem 1
\Biggr)
:
105
\sqrt{}
5 - 2
\surd
5
\pi
=
\infty \sum
k=0
\left[
1
2
,
1
5
,
1
5
,
3
5
,
4
5
,
1
10
1, 1, 1,
13
10
,
17
20
,
27
20
\right]
k
3750k3 + 2525k2 + 505k + 24
16k
.
Example 58
\Biggl(
1
2
,
1
2
,
1
2
,
2
5
in Theorem 1
\Biggr)
:
45
\sqrt{}
5 + 2
\surd
5
\pi
=
\infty \sum
k=0
\left[
1
2
,
1
5
,
2
5
,
2
5
,
3
5
,
7
10
1, 1, 1,
11
10
,
19
20
,
29
20
\right]
k
3750k3 + 2800k2 + 595k + 42
16k
.
Example 59
\Biggl(
1
2
,
1
2
,
1
2
,
3
5
in Theorem 1
\Biggr)
:
55
\sqrt{}
5 + 2
\surd
5
3\pi
=
\infty \sum
k=0
\left[
1
2
,
2
5
,
3
5
,
3
5
,
4
5
,
3
10
1, 1, 1,
9
10
,
21
20
,
31
20
\right]
k
1250k3 + 1025k2 + 215k + 16
16k
.
Example 60
\Biggl(
1
2
,
1
2
,
1
2
,
4
5
in Theorem 1
\Biggr)
:
195
\sqrt{}
5 - 2
\surd
5
\pi
=
\infty \sum
k=0
\left[
1
2
,
1
5
,
2
5
,
4
5
,
4
5
,
9
10
1, 1, 1,
7
10
,
23
20
,
33
20
\right]
k
3750k3 + 3350k2 + 655k + 36
16k
.
Example 61
\Biggl(
1
2
,
1
2
,
1
2
,
1
8
in Theorem 1
\Biggr)
:
480
\pi
\bigl( \surd
2 + 1
\bigr) =
\infty \sum
k=0
\left[
1
2
,
1
8
,
1
8
,
7
8
,
1
16
,
9
16
1, 1, 1,
11
8
,
13
16
,
21
16
\right]
k
15360k3 + 9920k2 + 1888k + 63
16k
.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
\pi -FORMULAE FROM DUAL SERIES OF THE DOUGALL THEOREM 1701
Example 62
\Biggl(
1
2
,
1
2
,
1
2
,
3
8
in Theorem 1
\Biggr)
:
224
3\pi
\bigl( \surd
2 - 1
\bigr) =
\infty \sum
k=0
\left[
1
2
,
3
8
,
3
8
,
5
8
,
3
16
,
11
16
1, 1, 1,
9
8
,
15
16
,
23
16
\right]
k
5120k3 + 3776k2 + 800k + 55
16k
.
Example 63
\Biggl(
1
2
,
1
2
,
1
2
,
5
8
in Theorem 1
\Biggr)
:
288
\pi
\bigl( \surd
2 - 1
\bigr) =
\infty \sum
k=0
\left[
1
2
,
3
8
,
5
8
,
5
8
,
5
16
,
13
16
1, 1, 1,
7
8
,
17
16
,
25
16
\right]
k
15360k3 + 12736k2 + 2656k + 195
16k
.
Example 64
\Biggl(
1
2
,
1
2
,
1
2
,
7
8
in Theorem 1
\Biggr)
:
1056
\pi (
\surd
2 + 1)
=
\infty \sum
k=0
\left[
1
2
,
1
8
,
7
8
,
7
8
,
7
16
,
15
16
1, 1, 1,
5
8
,
19
16
,
27
16
\right]
k
15360k3 + 14144k2 + 2656k + 105
16k
.
Example 65
\Biggl(
1
2
,
1
2
,
1
2
,
1
6
in Theorem 2
\Biggr)
:
10
\surd
3
\pi
=
\infty \sum
k=0
\left[ - 1
2
,
1
6
,
1
6
,
1
12
, - 5
12
1, 1, 1,
1
3
,
1
3
\right]
k
2160k3 - 372k2 + 68k + 5
16k
.
Example 66
\Biggl(
1
2
,
1
2
,
1
2
, - 1
3
in Theorem 2
\Biggr)
:
6
\surd
3
\pi
=
\infty \sum
k=0
\left[ - 1
2
,
1
3
, - 1
3
, - 1
3
, - 2
3
1, 1, 1,
1
12
,
7
12
\right]
k
135k3 - 48k2 - 7k + 2
16k
.
Example 67
\Biggl(
1
2
,
1
2
,
1
2
, - 1
4
in Theorem 2
\Biggr)
:
20
\pi
=
\infty \sum
k=0
\left[ - 1
2
,
1
4
, - 1
4
, - 1
8
, - 5
8
1, 1, 1,
1
8
,
5
8
\right]
k
960k3 - 232k2 - 38k + 5
16k
.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1702 W. CHU
Example 68
\Biggl(
1
2
,
1
2
,
1
6
,
1
2
in Theorem 2
\Biggr)
:
10
\surd
3
9\pi
=
\infty \sum
k=0
\left[
1
2
,
1
4
, - 1
4
,
5
6
, - 5
6
1, 1, 1,
1
3
,
2
3
\right]
k
k(120k2 - 26k + 5)
16k
.
Example 69
\Biggl(
1
2
,
1
2
,
7
6
, - 1
2
in Theorem 2
\Biggr)
:
6
\surd
3
\pi
=
\infty \sum
k=0
\left[ - 1
2
, - 1
4
, - 3
4
,
1
6
, - 1
6
1, 1, 1,
1
3
,
2
3
\right]
k
720k3 - 300k2 - 4k + 3
16k
.
Example 70
\Biggl(
1
2
,
1
2
, - 1
2
,
7
6
in Theorem 2
\Biggr)
:
27
\surd
3
\pi
=
\infty \sum
k=0
\left[ - 3
2
,
1
6
,
7
6
,
1
12
,
7
12
1, 1, 1,
1
3
, - 2
3
\right]
k
21 + 292k - 420k2 - 2160k3
16k
.
Example 71
\Biggl(
1
2
,
1
2
, - 1
2
,
1
4
in Theorem 2
\Biggr)
:
96
\pi
=
\infty \sum
k=0
\left[ - 3
2
,
1
4
,
3
4
,
1
8
, - 3
8
1, 1, 1,
3
8
, - 1
8
\right]
k
9 + 102k - 424k2 - 960k3
16k
.
Example 72
\Biggl(
1
2
,
1
2
, - 1
2
, - 1
4
in Theorem 2
\Biggr)
:
160
\pi
=
\infty \sum
k=0
\left[ - 3
2
,
5
4
, - 5
4
, - 1
8
, - 5
8
1, 1, 1,
1
8
, - 3
8
\right]
k
960k3 - 232k2 - 710k + 75
16k
.
Example 73
\Biggl(
1
2
,
1
2
, - 1
6
,
1
2
in Theorem 2
\Biggr)
:
28
3
\surd
3\pi
=
\infty \sum
k=0
\left[
1
2
,
1
4
, - 1
4
,
7
6
, - 7
6
,
13
12
1, 1, 1,
2
3
,
4
3
,
1
12
\right]
k
k(60k2 - 8k - 7)
16k
.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
\pi -FORMULAE FROM DUAL SERIES OF THE DOUGALL THEOREM 1703
Example 74
\Biggl(
1
2
,
1
2
, - 1
2
, - 1
3
in Theorem 2
\Biggr)
:
162
\surd
3
5\pi
=
\infty \sum
k=0
\left[ - 3
2
, - 1
3
, - 2
3
, - 4
3
,
4
3
1, 1, 1,
1
12
, - 5
12
\right]
k
135k3 - 48k2 - 106k + 24
16k
.
3.6. Series for \bfitpi .
Example 75
\Biggl(
5
2
, 2, 2,
3
4
in Theorem 2
\Biggr)
:
5\pi
16
=
\infty \sum
k=0
\left[ - 1
2
, - 1
4
,
1
8
, - 3
8
1
2
,
3
4
,
9
8
,
13
8
\right]
k
1 - 7k + 40k2
16k
.
Example 76
\Biggl(
5
2
, 2, 2,
1
4
in Theorem 2
\Biggr)
:
25\pi
16
=
\infty \sum
k=0
\left[ - 1
2
,
1
4
, - 1
8
, - 5
8
1
2
,
9
4
,
7
8
,
11
8
\right]
k
120k2 + 77k + 5
16k
.
Example 77
\Biggl(
3
2
, 2, 1,
1
4
in Theorem 2
\Biggr)
:
3\pi
8
=
\infty \sum
k=0
\left[ - 1
2
,
1
4
, - 1
8
,
3
8
1
2
,
5
4
,
7
8
,
11
8
\right]
k
1 + 11k + 106k2 + 240k3
16k
.
Example 78
\Biggl(
3
2
, 1, 1,
5
6
in Theorem 1
\Biggr)
:
36\pi
5
\surd
3
=
\infty \sum
k=0
\left[
1
2
,
1
3
,
2
3
,
1
6
,
5
6
,
14
9
3
2
,
5
3
,
5
4
,
7
4
,
7
6
,
5
9
\right]
k
60k2 + 64k + 13
16k
.
Example 79
\Biggl(
Chu and Zhang [21]:
3
2
, 1,
5
6
, 1 in Theorem 1
\Biggr)
:
20\pi
9
\surd
3
=
\infty \sum
k=0
\left[ 1,
1
3
,
2
3
,
1
4
,
3
4
,
8
5
3
2
,
3
2
,
3
2
,
3
5
,
7
6
,
11
6
\right]
k
12k2 + 15k + 4
16k
.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1704 W. CHU
Example 80
\Biggl(
1
2
, 1,
1
6
, 1 in Theorem 1
\Biggr)
:
20\pi
27
\surd
3
=
\infty \sum
k=0
\left[ 1,
1
3
,
2
3
,
3
4
,
5
4
1
2
,
1
2
,
3
2
,
7
6
,
11
6
\right]
k
4 - 11k - 69k2 - 60k3
16k
.
Example 81
\Biggl(
3
2
, 1,
5
6
, 2 in Theorem 1
\Biggr)
:
140\pi
27
\surd
3
=
\infty \sum
k=0
\left[ 2,
4
3
, - 1
3
,
3
4
,
5
4
1
2
,
3
2
,
3
2
,
11
6
,
13
6
\right]
k
60k3 + 133k2 + 85k + 13
16k
.
Example 82
\Biggl(
1
2
, 1, - 1
6
, 2 in Theorem 1
\Biggr)
:
700\pi
243
\surd
3
=
\infty \sum
k=0
\left[ 2, - 1
3
,
4
3
,
5
4
,
7
4
- 1
2
,
1
2
,
3
2
,
11
6
,
13
6
\right]
k
25 - 3k - 91k2 - 60k3
16k
.
Example 83
\Biggl(
3
2
, 1,
5
6
, 1 in Theorem 2
\Biggr)
:
4\pi
9
\surd
3
=
2
3
+
\infty \sum
k=1
\left[ 1,
2
3
, - 2
3
,
1
4
, - 1
4
1
2
,
1
2
,
3
2
,
5
6
,
7
6
\right]
k
20k2 + 7k + 2
16k
.
Example 84
\Biggl(
1
2
, 1,
1
6
, 1 in Theorem 2
\Biggr)
:
4\pi
81
\surd
3
=
2
3
+
\infty \sum
k=1
\left[ 1,
2
3
, - 2
3
,
1
4
,
3
4
1
2
,
1
2
, - 1
2
,
5
6
,
7
6
\right]
k
2 + 7k - 20k2
16k
.
Example 85
\Biggl(
5
2
, 1,
5
6
, 2 in Theorem 2
\Biggr)
:
10\pi
7
\surd
3
= 2 +
\infty \sum
k=1
\left[ 2,
5
3
, - 5
3
,
1
4
, - 1
4
1
2
,
1
2
,
5
2
,
7
6
,
11
6
\right]
k
12k2 + 17k + 8
16k
.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
\pi -FORMULAE FROM DUAL SERIES OF THE DOUGALL THEOREM 1705
Example 86
\Biggl(
3
2
, 1, 2,
5
6
in Theorem 2
\Biggr)
:
16\pi
3
\surd
3
= 20 +
\infty \sum
k=1
\left[ - 1
2
,
4
3
, - 4
3
,
1
6
,
5
6
1
2
,
5
3
,
1
4
,
3
4
,
7
6
\right]
k
56 - 33k - 270k2
16k
.
Example 87
\Biggl(
1
2
, 1,
1
6
, 2 in Theorem 2
\Biggr)
:
350\pi
243
\surd
3
= 30 +
\infty \sum
k=1
\left[ 2,
5
3
, - 5
3
,
3
4
,
5
4
1
2
,
1
2
, - 3
2
,
7
6
,
11
6
\right]
k
40 - k - 60k2
16k
.
Example 88
\Biggl(
3
2
, 1,
1
6
, 2 in Theorem 2
\Biggr)
:
32\pi
27
\surd
3
= - 8 +
\infty \sum
k=1
\left[ 2, - 2
3
, - 4
3
,
1
4
,
3
4
1
2
, - 1
2
,
3
2
,
5
6
,
7
6
\right]
k
k(15k + 2)(1 - 12k)
16k
.
Example 89
\Biggl(
5
2
, 1, 2,
5
6
in Theorem 1
\Biggr)
:
270\pi
7
\surd
3
=
\infty \sum
k=0
\left[
1
2
,
1
3
, - 1
3
,
1
6
,
5
6
5
2
,
8
3
,
5
4
,
7
4
,
7
6
\right]
k
70 + 409k + 627k2 + 270k3
16k
.
Example 90
\Biggl(
5
2
, 1, 2,
1
6
in Theorem 1
\Biggr)
:
756\pi
275
\surd
3
=
\infty \sum
k=0
\left[
1
2
, - 1
3
, - 2
3
,
1
6
, - 1
6
,
11
6
5
2
,
10
3
,
5
4
,
7
4
,
5
6
,
5
6
\right]
k
5 + 92k + 258k2 + 135k3
16k
.
3.7. BBP-series. In 1995, Simon Plouffe discovered the following amazing BBP-formula (named
after Bailey – Borwein – Plouffe [2] (Theorem 1)):
\pi =
\infty \sum
k=0
\Bigl( 1
16
\Bigr) k\biggl\{ 4
8k + 1
- 2
8k + 4
- 1
8k + 5
- 1
8k + 6
\biggr\}
that provides a digit-extraction algorithm for \pi in base 10. By decomposing the factorial fraction
in the summand into partial fractions, we can show that the next five series are all equivalent to the
above BBP-formula.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1706 W. CHU
Example 91
\Biggl(
3
2
, 1, 1,
3
4
in Theorem 1
\Biggr)
:
15\pi =
\infty \sum
k=0
\left[
1
2
,
3
4
,
1
8
,
5
8
3
2
,
7
4
,
9
8
,
13
8
\right]
k
120k2 + 151k + 47
16k
.
Example 92
\Biggl(
5
2
, 1, 2,
3
4
in Theorem 1
\Biggr)
:
63\pi
2
=
\infty \sum
k=0
\left[
1
2
,
3
4
,
1
8
, - 3
8
5
2
,
11
4
,
9
8
,
13
8
\right]
k
120k2 + 235k + 99
16k
.
Example 93
\Biggl(
3
2
, 1, 2, - 1
4
in Theorem 2
\Biggr)
:
21\pi
8
= 7 +
\infty \sum
k=1
\left[ - 1
2
, - 1
4
, - 3
8
, - 7
8
1
2
,
7
4
,
5
8
,
9
8
\right]
k
480k2 - 172k - 9
16k
.
Example 94
\Biggl(
5
2
, 1, 2,
3
4
in Theorem 2
\Biggr)
:
21\pi
10
= 7 +
\infty \sum
k=1
\left[ - 1
2
, - 1
4
, - 3
8
, - 7
8
5
2
,
3
4
,
5
8
,
9
8
\right]
k
23 + 10k - 240k2
16k
.
Example 95
\Biggl(
3
2
, 1, 2, - 5
4
in Theorem 2
\Biggr)
:
77\pi
8
= - 55
3
+
\infty \sum
k=1
\left[ - 1
2
, - 5
4
, - 7
8
, - 11
8
1
2
,
11
4
,
1
8
,
5
8
\right]
k
160k2 - 36k - 13
16k
.
There is another BBP-formula disguised in the article by Adamchik – Wagon [1]
2\pi =
\infty \sum
k=0
\Bigl( 1
16
\Bigr) k\biggl\{ 8
8k + 2
+
4
8k + 3
+
4
8k + 4
- 1
8k + 7
\biggr\}
.
Then the same approach of partial fractions can show that it has the following different infinite series
representations.
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
\pi -FORMULAE FROM DUAL SERIES OF THE DOUGALL THEOREM 1707
Example 96
\Biggl(
3
2
, 1, 1,
1
4
in Theorem 2
\Biggr)
:
5\pi
9
=
5
3
+
\infty \sum
k=1
\left[ - 1
2
,
1
4
, - 1
8
, - 5
8
3
2
,
5
4
,
3
8
,
7
8
\right]
k
7 - 6k - 80k2
16k
.
Example 97
\Biggl(
5
2
, 1, 2,
1
4
in Theorem 2
\Biggr)
:
15\pi
14
= 3 +
\infty \sum
k=1
\left[ - 1
2
,
1
4
, - 5
8
, - 9
8
5
2
,
9
4
,
3
8
,
7
8
\right]
k
19 - 62k - 80k2
16k
.
Example 98
\Biggl(
3
2
, 1, 2,
1
4
in Theorem 2
\Biggr)
:
15\pi
8
= 5 +
\infty \sum
k=1
\left[ - 1
2
, - 3
4
, - 1
8
, - 5
8
1
2
,
1
4
,
7
8
,
11
8
\right]
k
160k2 - 108k + 21
16k
.
Example 99
\Biggl(
3
2
, 1, 2, - 3
4
in Theorem 2
\Biggr)
:
45\pi
8
= 16 +
\infty \sum
k=0
\left[ - 1
2
, - 3
4
, - 5
8
, - 9
8
1
2
,
9
4
,
3
8
,
7
8
\right]
k
11 + 260k - 480k2
16k
.
Example 100
\Biggl(
3
2
, 1, 1,
5
4
in Theorem 1
\Biggr)
:
21\pi =
\infty \sum
k=0
\left[
1
2
,
1
4
,
3
8
,
7
8
3
2
,
5
4
,
11
8
,
15
8
\right]
k
65 + 413k + 812k2 + 480k3
16k
.
References
1. V. Adamchik, S. Wagon, \pi : A 2000-year search changes direction, Math. Educ. and Res., 5, № 1, 11 – 19 (1996).
2. D. Bailey et al., On the rapid computation of various polylogarithmic constants, Math. Comp., 66, № 218, 903 – 913
(1997).
3. W. N. Bailey, Generalized hypergeometric series, Cambridge Univ. Press, Cambridge (1935).
4. D. M. Bressoud, A matrix inverse, Proc. Amer. Math. Soc., 88, № 3, 446 – 448 (1983).
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
1708 W. CHU
5. Yu. A. Brychkov, Handbook of special functions: derivatives, integrals, series and other formulas, CRC Press, Boca
Raton, FL (2008).
6. L. Carlitz, Some inverse series relations, Duke Math. J., 40, 893 – 901 (1973).
7. X. Chen, W. Chu, Closed formulae for a class of terminating 3F2(4)-series, Integral Transforms and Spec. Funct.,
28, № 11, 825 – 837 (2017).
8. X. Chen, W. Chu, Terminating 3F2(4)-series extended with three integer parameters, J. Difference Equat. and Appl.,
24, № 8, 1346 – 1367 (2018).
9. W. Chu, Inversion techniques and combinatorial identities, Boll. Unione Mat. Ital., Ser. 7, 737 – 760 (1993).
10. W. Chu, Inversion techniques and combinatorial identities: strange evaluations of hypergeometric series, Pure Math.
and Appl., 4, № 4, 409 – 428 (1993).
11. W. Chu, A new proof for a terminating “strange” hypergeometric evaluation of Gasper and Rahman conjectured by
Gosper, C. R. Math. Acad. Sci. Paris, 318, 505 – 508 (1994).
12. W. Chu, Inversion techniques and combinatorial identities: a quick introduction to the hypergeometric evaluations,
Math. Appl., 283, 31 – 57 (1994).
13. W. Chu, Inversion techniques and combinatorial identities: basic hypergeometric identities, Publ. Math. Debrecen,
44, № 3/4, 301 – 320 (1994).
14. W. Chu, Inversion techniques and combinatorial identities: strange evaluations of basic hypergeometric series,
Compos. Math., 91, 121 – 144 (1994).
15. W. Chu, Inversion techniques and combinatorial identities: a unified treatment for the 7F6 -series identities, Collect.
Math., 45, № 1, 13 – 43 (1994).
16. W. Chu, Inversion techniques and combinatorial identities: Jackson’s q-analogue of the Dougall – Dixon theorem
and the dual formulae, Compos. Math., 95, 43 – 68 (1995).
17. W. Chu, Duplicate inverse series relations and hypergeometric evaluations with z = 1/4, Boll. Unione Mat. Ital.,
Ser. 8, 585 – 604 (2002).
18. W. Chu, Inversion techniques and combinatorial identities: balanced hypergeometric series, Rocky Mountain J.
Math., 32, № 2, 561 – 587 (2002).
19. W. Chu, q-Derivative operators and basic hypergeometric series, Results Math., 49, № 1-2, 25 – 44 (2006).
20. W. Chu, X. Wang, Summation formulae on Fox – Wright \Psi -functions, Integral Transforms and Spec. Funct., 19, № 8,
545 – 561 (2008).
21. W. Chu, W. Zhang, Accelerating Dougall’s 5F4 -sum and infinite series involving \pi , Math. Comp., 83, № 285,
475 – 512 (2014).
22. J. Dougall, On Vandermonde’s theorem, and some more general expansions, Proc. Edinb. Math. Soc., 25, 114 – 132
(1907).
23. I. Gessel, Finding identities with the WZ method, J. Symbolic Comput., 20, № 5-6, 537 – 566 (1995).
24. I. Gessel, D. Stanton, Strange evaluations of hypergeometric series, SIAM J. Math. Anal., 13, № 2, 295 – 308 (1982).
25. I. Gessel, D. Stanton, Applications of q-Lagrange inversion to basic hypergeometric series, Trans. Amer. Math. Soc.,
277, № 1, 173 – 201 (1983).
26. I. Gessel, D. Stanton, Another family of q-Lagrange inversion formulas, Rocky Mountain J. Math., 16, № 2, 373 – 384
(1986).
27. H. W. Gould, L. C. Hsu, Some new inverse series relations, Duke Math. J., 40, 885 – 891 (1973).
28. J. Guillera, About a new kind of Ramanujan-type series, Exp. Math., 12, № 4, 507 – 510 (2003).
29. J. Guillera, Generators of some Ramanujan formulas, Ramanujan J., 11, № 1, 41 – 48 (2006).
30. J. Guillera, Hypergeometric identities for 10 extended Ramanujan-type series, Ramanujan J., 15, № 2, 219 – 234
(2008).
31. E. D. Rainville, Special functions, The Macmillan Co., New York (1960).
32. S. Ramanujan, Modular equations and approximations to \pi , Quart. J. Math., 45, 350 – 372 (1914).
33. K. R. Stromberg, An introduction to classical real analysis, Wadsworth, INC, Belmont, California (1981).
Received 22.02.21
ISSN 1027-3190. Укр. мат. журн., 2022, т. 74, № 12
|
| id | umjimathkievua-article-6587 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:29:04Z |
| publishDate | 2023 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/1c/8fb866cd8611e0826ded63b6c4215d1c.pdf |
| spelling | umjimathkievua-article-65872023-02-05T14:53:26Z $\pi$-Formulae from dual series of the Dougall theorem $\pi$-Formulae from dual series of the Dougall theorem Chu, W. Chu, W. Classical hypergeometric series; The Dougall summation theorem; Gould-Hsu inverse series relations; Ramanujan's series for $1/\pi$; Guillera's series for $1/\pi^2$. special functions number theory UDC 517.5 By means of the extended Gould–Hsu inverse series relations, we find that the dual relation of Dougall's summation theorem for the well-poised $_7F_6$-series can be used to construct numerous interesting Ramanujan-like infinite-series expressions  for $\pi^{\pm1}$ and $\pi^{\pm2},$ including an elegant formula  for $\pi^{-2}$ due to Guillera. УДК 517.5 $\pi$-формули з дуальних рядів теореми Дугалла За допомогою узагальнених співвідношень  Гулда та  Хсу для  оберненого ряду доведено, що дуальне співвідношення теореми підсумовування Дугалла для добре збалансованого $_7F_6$-ряду можна використати для побудови багатьох цікавих виразів для нескінченного ряду, подібних до виразів, що були отримані  Рамануджаном для $\pi^{\pm1}$ і $\pi^{\pm2},$ включаючи елегантну формулу Гільєра для $\pi^{-2}$. Institute of Mathematics, NAS of Ukraine 2023-01-17 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/6587 10.37863/umzh.v74i12.6587 Ukrains’kyi Matematychnyi Zhurnal; Vol. 74 No. 12 (2022); 1686 - 1708 Український математичний журнал; Том 74 № 12 (2022); 1686 - 1708 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/6587/9345 Copyright (c) 2023 WENCHANG CHU |
| spellingShingle | Chu, W. Chu, W. $\pi$-Formulae from dual series of the Dougall theorem |
| title | $\pi$-Formulae from dual series of the Dougall theorem |
| title_alt | $\pi$-Formulae from dual series of the Dougall theorem |
| title_full | $\pi$-Formulae from dual series of the Dougall theorem |
| title_fullStr | $\pi$-Formulae from dual series of the Dougall theorem |
| title_full_unstemmed | $\pi$-Formulae from dual series of the Dougall theorem |
| title_short | $\pi$-Formulae from dual series of the Dougall theorem |
| title_sort | $\pi$-formulae from dual series of the dougall theorem |
| topic_facet | Classical hypergeometric series The Dougall summation theorem Gould-Hsu inverse series relations Ramanujan's series for $1/\pi$ Guillera's series for $1/\pi^2$. special functions number theory |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/6587 |
| work_keys_str_mv | AT chuw piformulaefromdualseriesofthedougalltheorem AT chuw piformulaefromdualseriesofthedougalltheorem |