An Algorithm for the Derivation of Rapidly Converging Infinite Series for Universal

Mathematical Constants

Cetin Hakimoglu-Brown tradelite@yahoo.com

1/27/09

Keywords: infinite series, hypergeometric function, pi formula, gamma function, universal constants, catalan’s constant, convergence improvement, binomial sums, beta function, pochhammer

Abstract

In this paper I’ll derive a novel algorithm that can be used to derive rapidly converging infinite series for various mathematical constants such as pi and values of the gamma function. A compendium of new

infinite series is given.

Introduction The purpose of this paper is lay out a novel, explicit algorithm that can be used to derive elegant, rapidly converging infinite series for various mathematical constants. New formulas for pi, gamma function, and other constants are derived inspired by David & Peter Borwein [2], Simon Plouffe, Travis Sherman [3], Christian Krattenthaler [4], and Boris Gour´evitch & Jes´us Guillera Goyanes [1], Bellard [6] who wrote similar papers based on deriving infinite series for constants. Examples of formulas that can be derived though the algorithm include:

( ) ( )

( )0

2 ! 130 10937 1160 12966 6

nn

n n

n nπ

=

∞

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

+=

−∑ (1.1)

And

( )( ) ( )

( )

31/2

1/30

2 1 102 597 3 1/ 3 3 213 192 28812 12

n n

nn

n n

n

π =

∞⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

+⋅ Γ=

⋅ −∑ (1.2)

1

These formulas were derived with the following integrals which we shall prove in this paper:

( )

( ) ( )( )( )

( ) ( )( )( ) ( )

1

0 0

1 12 11 1 1 2

bakn sn

s nk nk s n

a bz a b x xdx

a b z x x a b z=+

∞Γ

Γ Γ

+ +⋅ + + −=

+ + − − + +∑∫ (1.3)

We’ll also prove the more comprehensive integral-polynomial relation. This is the so called ‘machinery’ behind these infinite series:

( ) ( )( )

( ) ( )( )( ) ( )

( )

( ) ( )( )

( )( )

( ) ( )

( )

1

11 1

1 0

0

1 1

0 0

1 11 12 2

1 1

1

1 1

1...2

p p

p pg g

kn snn

nk s n

b ba a

sk

a g kn a g kna aa b g k s n a b g k s n

wa kna a

a b k s n

a ba bw

z a b a b z

Q x x x x xdx dx c

P xz x x

−

−= =

∞

=+

⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠

⎧ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎨⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

Γ Γ

Γ

+ + + ++ ++ + + + + + + + + +

=+ ++ +

+ + + +

+ ++ +=

⋅ + + + +

− −= =

− −

∏ ∏

∑

∫ ∫

⎫⎪ ⎪⎪ ⎪⎪ ⎪

⎬⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭

(1.4)

Derivation of Algorithm

We will begin by deriving formula (1.3) and apply it to derive the more complicated identity (1.4) Following the steps outlined in Boris Gour´evitch, Jes´us Guillera Goyanes’ paper ref [1], first consider the beta function:

( ) ( ) ( )( )( )

1

0

1 11

2sn bkn a kn a sn b

xk s n a b

x dx++ Γ Γ

Γ

+ + +−

+ + + +=∫

+

Then divide both sides by and take the summation nz

2

( ) ( ) ( )

( )( )0

1

0 0

1 112n n

n

sn bkn a

n s k z

kn a sn bx xdx

z a b n

∞

=

++

=

∞⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

Γ Γ

Γ + ⋅

+ + + +−=

+ + +∑ ∑∫

This is equal to:

( ) ( ) ( ) (( )( )

)0

1

0 0

1 111

2

nskba

nn n

x xx x

z s k z

kn a sn bdx

a b n

∞

= =

∞⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

Γ Γ

Γ

−−

+ ⋅

+ + + +=

+ + +∑ ∑∫ (2.0)

Summing an infinite geometric series as shown in ref [1]:

( ) ( ) ( )( )0

1 1

0 0

1 11

1

ns bk aba

skn

x x x xx x

z z x xdx z dx

∞

=

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

− −−

− −=∑ ∫ ∫ (2.1)

Using the identity: ( )( )( ) ( )nw w w nΓ = Γ + we derive:

( ) ( )( )( )

( ) ( ) ( ) ( )( )( ) ( )

1 1 1 1 12 22

kn sn

k s n

a b a ba b a b

kn a sn b

k s n a b +

Γ Γ Γ Γ

ΓΓ

1+ + +=

+ + + +

+ + + +

+ + + +

+ (2.2)

Plugging equations (2.1) and (2.2) into (2.0) and re-arranging terms we get equation (1.3) To make the integral useful for computing constants begin with a so called ‘seed integral’. Such an integral must be in the form:

( )( )

1

0

1 bax xdx c

P x−

=∫ where is a polynomial and is a constant ( )P x c

Then choose integer values of and k s such that the polynomial expansion of

( )1 skz x x− − is divisible by ( )P x . The value of is determined by computing the

quotient polynomial via polynomial long division such that the following identity is obtained:

z

( )Q x

( )( )

( ) ( )( )

1 1

00

1 1

1

b b

s

a a

kPx x

x xx Q xdx dx

z x− −

−=

−∫ ∫x

0

( )Q x can be generalized to: ( ) 1

1 1...p pp pa x a x a x aQ x −

−= + +

3

Multiplying ( )( )

1

0 1

1 ba

sk

x xdx

z x x−

− −∫ by ( )Q x and splitting the pa coefficients of

yields:

( )Q x

( )( )

( )( )

( )( )

( )( )

( )( )

11 110 0 1

1 01 11 00 0

1 1

1 1 1

1 1

1 1....

b ba p a p

p s sp bk k a

b ba a

s sk k

x x x xa dx a dx

z x x z x x x xdx

P xx x x xa dx a dx

z x x z x x

+ + −

−

+

⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭

− −+

− − − − −=

− −+

− − − −

∫ ∫∫

∫ ∫

Each of these …. split terms can be plugged into equation (1.3) : pa 0a

( )( )

( ) ( )( )

( ) ( )( )( ) ( )

( )( )

( ) ( )( )

( ) ( )( )( ) ( )

( )( )

( ) ( )

11

11

1

0 0

11

0 0

11

0

1 11 1121 2

11111 1

2 11

1

pp

pp

ba pkn sn

s nk nk s n

ba pkn sn

s nk nk s n

ba

sk

a

a

a

a p ba a p bx xdx

z a b pz x x a b p z

a p ba a p bx xdx

z a b pz x x a b p z

a a bx xdx

z a bz x x

−−

+

=+

+ −

=+

+

∞

∞

⋅

⋅

⋅

Γ Γ

Γ

Γ Γ

Γ

Γ Γ

Γ

+ + ++ + +−=

⋅ + + +− − + + +

+ ++ +−=

⋅ + + +− − + + +

+ +−=

⋅ +− −

∑∫

∑∫

∫

iii

( )( ) ( )

( )( ) ( )

( )( )

( ) ( )( )

( ) ( )( )( ) ( )

00

0

1

0 0

3

2 13

1 11 1121 2

kn snn

nk s n

bakn sn

s nk nk s n

a

a ba b z

a ba a bx xdx

z a bz x x a b z

=+

=+

∞

∞

⎧ ⎫⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎨ ⎬⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪⎪ ⎪

⋅⎪ ⎪⎪ ⎪⎪ ⎪⎩ ⎭

Γ Γ

Γ

+ ++ + +

+ ++ +−=

⋅ + +− − + +

∑

∑∫(2.3)

4

Also note the following pochhammer and gamma function identities:

( ) ( ) ( )( )

( ) ( )( )

( ) ( ) ( )

( ) ( ) ( )

1

1

1

1

12 2

1

1 1

2 1 2

1 1

p

k s ng

p

kn kng

p

g

p

g

k s na b g k s n

a b p a ba b g

a g kna p aa g

a b p a b g a b

a p a g a

⎛ ⎞⎜ ⎟⎝ ⎠

+=

=

=

=

+

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

Γ Γ

Γ Γ

+ + + + ++ + + = + +

+ + +

+ ++ + = ++

+ + + = + + + + +

+ + = + +

∏

∏

∏

∏

(2.4)

Putting it all together utilizing (2.3) and (2.4) gives:

( ) ( )( )

( ) ( )( )( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( )

( ) ( )( ) ( )

1

1 1

0

11 1

11 2 2

1

1 11 1

2

1 1

2p

p

p kn sng

p pn

k s ng g

kn snn

nk s n

p

g

a g kna a ba g

a b g k s nz a b g a b a b z

a b g

a p ba a p bz a b p a b p z

a g a b=

+= =

=+

=

∞⋅=

⎛ ⎞ ⎛ ⎞⎛ ⎞⎛ ⎞+ +⎜ ⎟ + +⎜ ⎜ ⎟⎜ ⎟⎜ ⎟ +⎝ ⎠⎜⎝ ⎠ ⎝ ⎠⎜⎛ ⎞ ⎛ ⎞+ + + + +⎛ ⎞⎜⋅ + + + Γ + + + +⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎜ + + +⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠

Γ Γ

Γ

Γ Γ

+ + ++ + +⋅ + + + + + +

+ + + ∏

∏ ∏

∑

∏0n=

∞ ⎟⎟⎟⎟⎟

∑

Also note how the ( )1

1p

ga b g

=+ + +∏ and (

1)

p

ga g

=+∏ are eliminated such that we

simplify and obtain:

5

( ) ( )( )

( ) ( ) ( )

( )( ) ( )( ) ( )

( )( )

1

1

0

1

0

1 11 12

1 2

1

1

p

kn snp g

pn

k s ng

p

n

ba p

sk

a g kn a baz a b

a b g k s n a b z

a

a b

x xdx

z x x

=

+=

=

+

∞

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

+ + + +Γ Γ

=⋅Γ + +

+ + + + + + +

+ +

−− −

∏

∏∑

∫

Performing this procedure for pa … , summing the p+1 terms, and factoring out 0a

( ) ( )( )( ) ( )

1 1

2kn sn

nk s n

a ba b z

+

+ +

+ + gives equation (1.4), which completes the proof.

One final pochhammer identity that will be encountered frequently in this paper is

( )1

0

kkn

nky n

a ya kk

−

=

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠

+= ∏ (2.5)

Another forumula that will be used extensively later is the Euler integral:

{ } ( )( ) ( )

( )( )

1 22

1

111

2 1 21 2 1

1

0

12 1

1, ; ;

y xx

x

y x xx y x zx

F x x y z dx− −− −

=Γ Γ − −

Γ∫ (2.6)

Deriving infinite series for a constant (c) is as simple as plugging in values of k, s, a, b, z and the pa coefficients of into formula (1.4). ( )Q x

Deriving Formulas for Pi We shall use algorithm (1.4) to find various rapidly converging formulas for pi. We’ll begin by deriving formula (1.1) Consider a hypergeometric function for arcsine converted into integral form via (2.6)

( ) ( )1 12 1

22 2

02

sin1 3 12 1 1, ; ; 12 2 1 2

11

ww xF ww w x w

w

− −⎛ ⎞⎧ ⎫= − =⎜ ⎟⎨ ⎬− ⎛ ⎞⋅⎩ ⎭ ⎝ ⎠ − ⎜ ⎟−⎝ ⎠

∫/2

dx (2.7)

6

Then let w=1/2 so that 1 1/2

0

33 1

3

x dxxπ −

=+

∫

To derive a rapidly converging infinite series for this integral using (1.4) let k=1 and s=2.

Then let a=-1/2 b=0 and ( ) 13xP x = + and find Q(x) such that:

( )

( )

1/21 1 1/2

20 01 1

3

x Q x xdx dxxz x x

− −

=− − +

∫ ∫

Expand and perform polynomial division ( 21z x x− − ) 3 21 23x x x x z+ − + − +

A value of z = -48 and is obtained. Thus: ( ) 23 15 4Q x x x= − + − 8

( )

( )

1/2 21 1 1/2

20 0

5 16 13 91 1

3

x x x xdx dxxz x xπ

− −− + −= =

− − +∫ ∫

3− (2.8)

Then plug the following values into (1.4):

a=-1/2, b=0, k=1, s=2, z=-48, 0 16a = , 1 5a = − , 2 1a = and obtain:

( ) ( )( )

( ) ( )( ) ( )

2

0 3

1 21 2 13 1 2 116 59 48 3 2 3 2 3 3 2 3 5 2 33 2

148

n nn

n n

n nn n

π=

∞ ⎛ ⎞Γ Γ ⎛ ⎞ ⎛ ⎞⎛+ + +− +⎜ ⎟⎜ ⎟ ⎜ ⎟⎜− Γ + + +⎝ ⎠ ⎝ ⎠⎝⎝ ⎠

− =−

∑ 2 3 2 nn⎞⎟⎠

Using identity (2.5) and simplification on the equation above, formula (1.1) is derived:

( ) ( )( )0

2 ! 130 10937 1160 12966 6

nn

n n

n nπ

=

∞

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

+=

−∑

Interestingly, formula (1.1) converges at a rate of log(324) or 2.5 digits per term using an arcsine formula for ½, whereas the original arcsine formulas utilizing the binomial theorem converge at best at a rate of only log(4) for ½. Yet formula (1.1) is compact, unique, and converges about four times faster.

7

A faster formula can be obtained that adds five digits per term. Let s=4, k=2, a=-1/2, and

b=0. ( ) 13xP x ⎛= +⎜

⎝ ⎠⎞⎟ Then expand and performing division for Q(x) the

following identity is obtained:

( )42 1z x x− −

( )

( )

1/2 5 4 3 21 1 1/2

420 0

7 27 85 256 768 1 33 92304 1 1

3

x x x x x x xdx dxxx xπ

− −− + − + − += =

− − +∫ ∫ (2.9)

After some labor a pi formula derived using (1.4), in ‘BBP’ notation [2]:

( ) ( )

5

2

10

4 ! 6 !3 127169 1070 131 26 (2 )!(12 )!(9) 12 1 12 5 12 7 12 11n

n

n nn n n n n n

π +=

∞ ⎛ ⎞⎡ ⎤ ⎛ ⎞⎣ ⎦⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎜ ⎟

⎝ ⎠

= − −+ + + +∑ + (2.10)

Or using identity (2.5) we can express (2.10) in pochhammer notation:

( ) ( )

( ) ( ) ( ) ( ) ( )

3 2

4 10

4 ! 671840 1289936 782458 15083531155 72 13 12 17 12 19 12 23 12n

n n n nn

n n n nπ +

=

∞ + + += ∑ (2.11)

To derive another pi formula let s=2, k=2, a=-1/2, b=0 and ( ) 13xP x = + to obtain:

( )

( )

1/2 31

220

5 16 48 39144 1

x x x xdx

x xπ

− − + − +=

− −∫ (2.12)

After some labor using (1.4) we get the BBP form equation below which ads 3.4 digits/term:

10

03 1 5717 413 45 52

8 8 1 8 3 8 5 8 79

4nn n n n n n

n

π=

∞ ⎛ ⎞⎜ ⎟⎛ ⎞ ⎝ ⎠

⎜ ⎟⎝ ⎠

− − ++ + + +

=∑ (2.13)

Using Multivariable Values of P(x) and Q(x) to Derive Infinite

Series for Pi

A slew of binomial identities for pi and other constants can be found using equations (1.3) and (1.4), but extending it to multiple variables. In this section Q(x) becomes Q(x,w), P(x) becomes P(x,w), and z becomes a function of w.

8

To begin, using equation (1.3) let a=0, b=0, k=1, s=1, z=w to obtain the identity:

( )

1

0 10

12(1 )

2 1n ndx

nw x xw n

n

∞

= + ⎛ ⎞⎜ ⎟⎝ ⎠

=− −

1

+∑∫ (3.1)

By evaluating the left hand integral we see that (3.1) is equal to [3]:

( )14 tan 1 4 1

4 1

w

w

− −

− (3.2)

Do derive a faster series for (3.1) let k=3, s=3, a=0,b=0, z=z, and expand ( )33 1z x x− −And let P(x,w) = 2x x w− + . Performing polynomial division, we need to find Q(x,w) and

a new value of z in terms of w. ( ) ( )2 6 5 4 33 3x x w x x x x z− + − + − +

We find: Therefore: ( ) ( )4 3 2 2, 2 (1 ) ,Q x w x x w x xw w z w= − + − + + = 3

( )4 3 2 21

3 3 30

1

0

2 (1 ) 1(1 ) 1( )

x x w x xw wdx dx

w x x w x x− + − + +

=− − − −∫ ∫ (3.3)

With regards to equation (3.1) we can let 2w w−= ,divide by w, and take the derivative in terms of w such that:

( )

( )( )

1 22

2 3/22

2 1 2

0 0

4 tan 4

42 2 42 1

n n

n n

ddw

ww w

w w ww wn nn

n n

−∞ ∞+

= =⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

= = +− −

−

+∑ ∑ (3.4)

An arbitrary number of integrals and derivatives of (3.4) can be taken in terms of w to obtain various central binomial identities. Applying formula (1.4) on (3.3), and after some labor obtain: (The variable r denotes the powers of n that results from taking multiple derivatives)

( )

( ) ( )( )( )

( )( ) ( )

2 22 32

1

14

2

9 9 2 2 3 12 4263 2 3 2 6 3 6 1 6 1 3 1 33

r

n

r rrn

n

n n

w n n w nw wwn n n n n n nn

wn

n∞

=

∞

= ⎛ ⎞⎜ ⎟⎝ ⎠

⎛ ⎞⎜ ⎟+⎜ ⎟⎛ ⎞⎝ ⎠⎜ ⎟

⎝ ⎠

=

+ + ++ +

⋅ + + + + +∑

∑

r+

(3.5)

9

To derive an elegant pi formula let r=0 and 2w w= in (3.5). Then divide (3.5) by w and take multiple derivatives such that the denominators vanish, ensuring you divide (3.5) by w each time before the derivative is taken. Then finally let w=1 to obtain the identity:

2 2

1 1 1

27 43 16 32 1263 16 2 23

n n n

nn n nn n nn n n

∞ ∞ ∞

= = =

− += − +

⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

∑ ∑ ∑1 2n n

n

∞

=∑ (3.6)

Using (3.4) and taking multiple derivatives we arrive at:

2

1

27 4 40 3381

6363

n

nnnn

π∞

=

− + 4= +⎛ ⎞⎜ ⎟⎝ ⎠

∑ (3.7)

A faster converting series can be obtained by letting k=5, s=5 such that

( )( )

( )

( )( ) ( ) ( )

( ) ( )

5

8 7 6 5 2 4

2 3 2 3 2 3 4

1

5 50

1

0

,

, (1 )

4 6 3 4 3 1,

2

1(1 ) ,

x w

P x w w x x

x x w x w x w w xQ x w

w w x w w x w x w

Qdx dx

w x x P x w

= − −

⎛ ⎞− + − + − + − +⎜ ⎟=⎜ ⎟+ − + − + +⎝ ⎠

=− −∫ ∫

(3.8)

Using formula (1.4) and after some labor the identity is obtained:

4 3 2

4 3 2

0

0

213125 278000 139975 26800 1596

16 128 344 352 105 1120 316 172881

105

2

n

n

n n n n

n n n n

nn

nn

π

∞

=

∞

=

− + − +=

⎛ ⎞⎜ ⎟⎝ ⎠

− + − += +

⎛ ⎞⎜ ⎟⎝ ⎠

∑

∑

(3.9)

Rapidly Converging Infinite Series for the Gamma Function

Elegant, rapidly converging infinite series for the gamma function such as (1.2) can be derived using identities (1.3) and (1.4). In this section we’ll also prove an ‘accidental’

10

quadratic hypergeometric transformation that is a byproduct of this method, as well as other identities. Begin with Kummer’s formula [5] and express it in integral form via (2.6):

{ } ( ) ( )( ) ( )

( )( ) ( )

( )1 211

0

2 3 2 22 1 1, ;2 ; 1

3 2 2 2 211

hhh hF h h

h h hx x

dxx

−−Γ − Γ Γ −− − = =

Γ − Γ Γ − Γ

−+∫ (4.1)

Since we can let k =1, s =1, z = -2 and ( ) ( )(2 1 2 1x x x x− − − = − + ) ( ) 2Q x x= − And after isolating the integral in (4.1) we have the identity:

( ) ( )

( )( ) ( ) ( )

( )

1 21 1 2 11 1

0 0

2 2 22 3 2

1 112 1

hh h hx h hh

x x x xdx dx

xx xπ

−− − −− − ΓΓ −

Γ− −= =

+− − −∫ ∫ (4.2)

The leftmost integral in (4.3) can be evaluated using (1.4). setting 0 2a = − 1 1a =a=h-1 b=1-2h :

( ) ( )( )

( ) ( )( ) ( )

( ) ( )( )0 2

2 2 2 22

2 2 2 2 2 3 22

2 22

n nn

n n

hh h h hh nh h nh

h π

=

∞Γ Γ − Γ Γ −+⎛ ⎞− =⎜ ⎟− Γ − − + Γ −⎝ ⎠−

−

−∑ h

After some labor and using (2.5) we obtain this quickly converging formula for computing values of the gamma function:

( ) ( )

( )( ) ( ) ( )( ) ( ) ( )

22 1

0

1 2 3 3 11 22 8 2 1 2 3 2

hn n

nn nn

h h hh

h h n hh h

−

=

∞⎡ ⎤⎣ ⎦+ Γ

Γ

− + +=

− + +∑ (4.3)

To derive (1.2) let h=1/3 in equation (4.3) and simplify:

( ) ( ) ( ) (( )

)( ) ( )

1/2 32/3

0

1 3 2 3 2 3

7 6 5 3

3 2 3 29 8

n nn

n nn

nπ =

∞⎡ ⎤⎣ ⎦Γ +=

−∑

But ( ) ( ) ( )2 3 3 2 2 5 3n n+ = n so we can write:

( ) ( )

( ) ( )

1/2 32/3

0

1 3 2 3 2 7 12 1 1, ; ;3 6 87 6

3 218 8

nn

nnF

π =

∞⎡ ⎤ −⎧ ⎫⎣ ⎦ = ⎨ ⎬⎩ ⎭

Γ=

−∑ (4.4)

(4.4) happens to also be a quadratically transformed hypergeometric function, which via (2.6) can be converted to an integral:

11

( ) ( )( ) ( )

( ) 1/21/2 32/3 1/31

0

1 3 7 61 2 2 3

3 2 118 1 8/

x xdx

xπ

−−⎡ ⎤⎣ ⎦Γ ⋅Γ

Γ Γ −=

+∫

After some labor we obtain the simplification:

( ) 1/2

1/2 1/31

0

14 39 1 8/

x xdx

xπ

−−⋅⋅ −

=+∫ (4.5)

Letting

( ) ( )( ) ( ) 1 0

1, 1, 1/ 3, 1/ 2, 1 / 8 ,

8 72 , 72, 8, 72

k s a b P x x

Q x x z a a

= = = − = − = +

= − = − = = −

We arrive at the identity:

( ) ( ) ( )( )

1/2 1/21/3 1/31 1

0 0

8 721

1 11 8 72/

xx

x x x xdx dx

x x

− −− − −

−

− −=

+ − −∫ ∫

Plugging the above values into (1.4) and simplifying through (2.5) we get (1.2):

( )( ) ( )

( )

31/2

1/30

2 1 102 597 3 1/ 3 3 213 192 28812 12

n n

nn

n n

n

π =

∞⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

+⋅ Γ=

⋅ −∑

An even faster formula for (1 3Γ ) can be found that gives 3.6 digits/term though the following integral identity where we let s=2 and k=1:

( ) ( ) ( )( )

1/21/2 2

2

1/31/31 1

0 0

10 811 38 8 1

111 648/

x x

x

x xx xdx dx

x xπ

−− −− − + −

−

−−= =

+ − −∫ ∫ 18

Applying (1.4) and (2.5) and after some labor obtain:

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

2

1/3

33/2

0

2 3 1 4 3 4 8400 9248 22971 3

19 18 25 18 31 18 4374

91 32

n n nn

n n nn

nnπ =

∞⎡ ⎤ +⎣ ⎦⋅ Γ +=

−∑ (4.6)

12

Higher Order Constants & the Limitations of Accelerating the Generalized Hypergeometric Function

There series we’ve been deriving so far involve constants that can be expressed though the hypergeometric function of the form { }2 1 1, ; ;F x y z . Such constants involve logarithms, root extractions, the gamma function, and inverse trigonometric functions. Higher order constants such as Catalans’s Constant, Zeta 3, etc can’t be expressed through { }2 1 1, ; ;F x y z , but require a more generalized hypergeometric function of the

form: { }1 1 11, ,..., ; ;...; ;q q q qF x x y y z+ . Unfortunately, the algorithm (1.4) isn’t as efficient at generating rapidly converging series for constants that are only expressible though a hypergeometric function where q>1. To the best of my knowledge for (1.4) to work on higher order constants such as Zeta 3, the value of ‘ s ’ must be set to zero. Accelerating the convergence of a q>1 hypergeometric function via (1.4) by setting s=0 equates to taking multiple terms of

{ }1 1 1 ...;1, ,..., ; ; ;q q q qF x x y y z+ ‘k’ terms at a time. So if k=2, the series generated by (1.4)

would converge at a rate of . 2z In this next example we will show how to accelerate the convergence of a hypergeometric function of the form { }1 2 1 23 2 1, , ; ; ;F x x y y z using (1.4), and then apply it

to derive faster converging infinite series for Catalan’s Constant and ( )1 4Γ . Begin with the integrals:

{ } ( )( ) ( ) ( ) { }22

2 2 2

2 211 2 1 2 1 1

1 1

03 22 1, , ; ; ; 1 1 1, ; ;y xxy

x y xF x x y y z F x y xzx x d− − −

=Γ Γ −

Γ−∫ x

(5.1)

{ } ( )( ) ( )

( ) 111

1 1 1

11 1

1 1

1

0

12

11 1, ; ;

y xxy y yx y x yxz

F x y xz dy− −−−

=Γ Γ − −

Γ∫ (5.2)

Using the integral (5.2), choosing the values below for k and s, and performing polynomial long division we plug the following into (1.4):

( ) ( )

( ) ( ) ( )

1 1 1 2 2

2 2

10, 2, 1 , 1 ,

1, , 1 , , ,

s k b y x a x zx z

yP x y z xyz Q x y zxz x z

= = = − − = − =

⎛ ⎞= − = +⎜ ⎟⎝ ⎠

And obtain:

13

( )( )

( ) ( )

( ) ( )( )

( ) ( )( ) ( )

1 11 1

22 2

2 211 1 1 1

2 21 1 1

1 11 11 1

2

1 1

0 0

2

20

1 11, ,

2 12

n

y y

n

n

, ,x xx x

n

y y y y Q x y zP x y z y

x z

x xzx z x y x x ny y xz y n

dy dy− −− −− −

=

∞

− −

−

⎛ ⎞⋅Γ Γ − ++⎜ ⎟⎜ ⎟Γ +⎝ ⎠x z

= =∫ ∫

∑ (5.3)

Plugging (5.3) into (5.2), and then plugging that result into (5.1) and simplifying we get:

( )( ) ( ) ( ) ( ) ( )

( )( )

{ }

2 12 1

2 2 2 1 1

2 22

1

1 2 1 2

1 1 20

20

21

2

3

1

2 1, , ; ; ;

ny x n

n

x

n

x xzy xdx

x y x y y n

F x x y y z

x x− − −

=

z x n∞Γ +⎛ ⎞+⎜ ⎟Γ Γ − +⎝ ⎠

=

−∫ ∑ (5.4)

Then we extract and convert the following integrals from (5.4) into pochhammer & gamma notation via the beta function:

( ) ( ) ( )( )

( ) ( ) ( )( ) ( )

( ) ( ) ( )( )

( )( ) ( ) ( )( )( ) ( )

2

2

2 2 2 22 2 2 2

2 2 22

2 2 2 22 2 2 2

2 2 2 2

2 2

2 2

11 1

0

1 1

0

2

2

22

22 12 1 2 1

1

1

n

n

n

n

y x

y x

n x

n x

x x y xn x y xdx

n y y y

2

2

x n x x y xn x y xdx

n y y n y y

x x

x x

− − −

− −

+

+

Γ Γ −Γ + Γ −= =

Γ + Γ

+ Γ Γ −Γ + + Γ −= =

Γ + + + + Γ

−

−

∫

∫

(5.5)

Then plugging (5.5) back into (5.4) and simplifying gives the final result:

{ } ( ) ( ) ( )( ) ( )

( )( )( )( )

1 2 1 2

1 2 1 2

2

1 2 1 22 2

2 20

2 23 2 1

2 21, , ; ; ;

nn n

n nn

x x z z x n x ny y y n y n

F x x y y z=

∞ ⎛ ⎞+ += +⎜ ⎟⎜ ⎟+ +⎝ ⎠∑ (5.6)

The procedure used to derive (5.6) can be generalized:

{ } ( )( )

21 1 1

1 1 21

21, ,..., ; ;...; ; 1

2

q qgg n

q q q qg gg g nn

xx nF x x y y z z

y n y+= ==

∞ ⎛ ⎞⎛ ⎞⎛ ⎞+⎜ ⎟= +⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟+⎝ ⎠⎝ ⎠⎝ ⎠

∏ ∏∑ (5.7)

With regards to Catalan’s Constant it is well-known that

14

( )( ) ( )

( )220 0

11 3 3 1 83 2 ln 2 322 2 2 4 3 3 2 12 1

11,1, ; ; ;n

n nn nnn

F π

= =

∞ ∞ −⎧ ⎫ = = − +⎨ ⎬⎛ ⎞ +⎩ ⎭ + ⎜ ⎟⎝ ⎠

∑ ∑ (5.8)

Plugging the hyqpergeometric terms of (5.8) into (5.6) and simplifying gives an infinite series for (5.8) that converges twice as fast:

( ) ( )( )( )

2

220 0

54 192

2 1 4 1 4 32

1 40n n

nn

n n nn n

n= =

∞ ∞ + +=

⎛ ⎞ ⎛ ⎞+ + +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

∑ ∑ 4n (5.9)

Using a transformation on (5.2) letting k=3 gives:

( ) ( )( )( )( )

4 3 2

220 0

17172 15903 6405 95642 6

2 1 6 1 6 3 6 53

1 6804n n

n n nn n

n n n nn n

n= =

∞ ∞ + + + +=

⎛ ⎞ ⎛ ⎞+ + + +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

∑ ∑ (5.10)

In addition, rapidly converging formulas for ( )1 4Γ can be derived using (5.6) based on accelerating a quadratic transformation. Begin with the following quadratically transformed hypergeometric functions:

( ) 21/4

1 1 12 1 , ;1;4 4 8 3 42

F π−⎧ ⎫ =⎨ ⎬⎩ ⎭ ⎡ ⎤⎣ ⎦Γ

(5.11)

( )2

23 1 41 3 12 1 , ;1;

4 4 9 4F

π

π

⎡ ⎤⎧ ⎫ ⎣ ⎦=⎨ ⎬⎩ ⎭

Γ (5.12)

(Note: Unlike equation (4.4), (5.11) and (5.12) can’t be derived with (1.4). Proofs regarding quadratic hypergeometric transformations is given in [7].)

However, (5.11) is equal to 1 1 13 2 1, , ;1;1;4 4 8

F −⎧ ⎫⎨ ⎬⎩ ⎭

and (5.12) can be written as

1 3 13 2 1, , ;1;1;4 4 9

F ⎧⎨⎩ ⎭

⎫⎬ . Plugging these values into (5.6) and simplifying via (2.5) gives

( ) ( )( )

( ) ( )

2

23/2 4

2

0

36 3 1 4 640 608 147 8 !

2 1 ! 4 !24 nn

n n n

n nπ =

∞⎡ ⎤ + +⎣ ⎦ =+⎡ ⎤⎣ ⎦

Γ∑ (5.13)

15

( )( ) ( ) ( )

( )

21/2

1/4

22

0

1 8 5 81282 2 1 !2 3 4

448 448 127 n nn

n nn nπ

=

∞ ⎛ ⎞= ⎜⎜ +⎡ ⎤ ⎝⎣ ⎦

+ +Γ

∑ ⎟⎟⎠ (5.14)

The formulas (5.13) and (5.14) conclude the paper. So far, we’ve proven and

showed examples how implementing algorithm (1.4) involving polynomial division can be used to derive elegant, rapidly converging infinite series for various mathematical constants, as well as accelerate the convergence of existing series as in examples (5.13, 5.14, 5.9, and 5.10). To the best of my knowledge, most of the series derived in this paper are original, although the procedure involving polynomial division isn’t. However, I have yet to see an explicit, closed form algorithms like (1.3) or (1.4) for deriving a wide spectrum of infinite series.

An open problem related to this paper is if there exists a transformation using

(1.4) or a variant of it for { }1 1 11, ,..., ; ;...; ;q q q qF x x y y z+ q>1 where s >0. My best attempts to derive closed form, transformation for non trivial 3F2 or higher hypergeometric functions for s>0 have failed. By non trivial what I mean is that n nx y− cannot be a positive integer. { }3 2 1 4 5 2 3 1 3, , ; ; ;F z , for example, is trivial because 4-3 is an integer.

{ }3 2 1 3 5 2 4 1 3, , ; ; ;F z , on the other hand, isn’t trivial because 3-4 is a negative integer. Also, 5/2-1/3 isn’t an integer.

I would like to thank the Borwein brothers, Simon Plouffe and Boris Gour´evitch & Jes´us Guillera Goyanes for their work, which inspired me to write this paper.

References:

[1] Boris Gour´evitch, Jes´us Guillera Goyanes, Construction Of Binomial Sums For pi And Polylogarithmic Constants Inspired By BBP Formulas (2006). [2] D. Bailey, P. Borwein and S. Plouffe, On The Rapid Computation of Various Polylogarithmic Constants, Math. Comp., 66(1997), 903–913. [3] Travis Sherman, Summation of Glaisher- and Apéry-like Series, University of Arizona (2000),http://math.arizona.edu/~ura/001/sherman.travis/series.pdf [4] G. Almkvist, C. Krattenthaler and J. Petersson, Some new formulas for _, preprint, Matematiska Institutionen, Lunds Universitet, Sweden. and Institut fur Mathematik der Universitat Wien, Austria (2001), http://www.arxiv.org/pdf/math.NT/0110238

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[5] "List of hypergeometric identities" Wikipedia, The Free Encyclopedia. Wikimedia Foundation, Inc. 10 Aug. 2004. http://en.wikipedia.org/wiki/List_of_hypergeometric_identities [6] Fabrice Bellard, _ page, http://fabrice.bellard.free.fr/pi/. [7] Earl D. Rainville, Ph.D. “Special Functions”, page 65, The MacMillan Company, NY., 1960.

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