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Article

# Further Closed Formulae of Exotic 3F2-Series

by
Wenchang Chu
1,2
1
School of Mathematics and Statistics, Zhoukou Normal University, Zhoukou 466001, China
2
Department of Mathematics and Physics, University of Salento, 73100 Lecce, Italy
Axioms 2023, 12(3), 291; https://doi.org/10.3390/axioms12030291
Submission received: 20 December 2022 / Revised: 24 February 2023 / Accepted: 9 March 2023 / Published: 10 March 2023
(This article belongs to the Special Issue 10th Anniversary of Axioms: Mathematical Analysis)

## Abstract

:
By making use of the linearization method, we examine a class of nonterminating $3 F 2$-series with five free integer parameters that yields twenty summation formulae. Under the Kummer and Thomae transformations, six classes of exotic $3 F 2$-series are consequently evaluated in closed forms. There are overall 100 identities recorded in the present paper
MSC:
Primary 33C20; Secondary 33F10

## 1. Introduction and Outline

Denote by $N$ and $Z$, respectively, the sets of natural numbers and integers with $N 0 = N ∪ { 0 }$. The shifted factorials are given by $( x ) 0 = x 0 = ≡ 1$ and
$( x ) n = x ( x + 1 ) ⋯ ( x + n − 1 ) x n = x ( x − 1 ) ⋯ ( x − n + 1 ) for n ∈ N .$
We can express them, even when $n ∈ Z$, as the quotients
$( x ) n = Γ ( x + n ) Γ ( x ) and x n = Γ ( 1 + x ) Γ ( 1 + x − n ) ,$
where the $Γ$-function is defined by the Euler integral
$Γ ( x ) = ∫ 0 ∞ u x − 1 e − u d u for ℜ ( x ) > 0 .$
For brevity, their fractional forms are concisely shortened as
$α , β , ⋯ , γ A , B , ⋯ , C n = ( α ) n ( β ) n ⋯ ( γ ) n ( A ) n ( B ) n ⋯ ( C ) n , Γ α , β , ⋯ , γ A , B , ⋯ , C = Γ ( α ) Γ ( β ) ⋯ Γ ( γ ) Γ ( A ) Γ ( B ) ⋯ Γ ( C ) .$
According to Bailey [1], the generalized hypergeometric series is defined by
$1 + p F p a 0 , a 1 , ⋯ , a p b 1 , ⋯ , b p | z = ∑ n = 0 ∞ ( a 0 ) n ( a 1 ) n ⋯ ( a p ) n n ! ( b 1 ) n ⋯ ( b p ) n z n .$
When $z = 1$, this series is convergent only if the “parameter excess” (i.e., the difference between the sum of the denominator parameters and that of the numerator ones) has a positive real part.
There exist many strange evaluations of hypergeometric series (cf. [2,3,4,5,6,7,8] for example). Recently, Campbell, D’Aurizio and Sondow [9,10] discovered two mysterious-looking formulae (see D1 and D12)
$3 F 2 1 4 , 3 4 , 1 2 1 , 3 2 | 1 = 4 ln ( 1 + 2 ) π , 3 F 2 1 4 , 3 4 , − 1 2 1 , 1 2 | 1 = 2 + ln ( 1 + 2 ) π .$
Campbell and Abrarov [11] found, among the others, the following two further ones (see F10 and G8)
$3 F 2 3 2 , 3 4 , − 1 4 1 , 7 4 | 1 = 3 π 3 2 − log ( 1 + 2 ) 2 Γ ( 1 4 ) 2 , 3 F 2 3 2 , 1 4 , 5 4 1 , 9 4 | 1 = 5 π 3 2 − log ( 1 + 2 ) 8 Γ ( 3 4 ) 2 .$
These series are said “exotic” because one numerator parameter minus a denominator parameter results in a negative integer. By examining carefully these seemingly unrelated series, we find that they are connected, under the Thomae and Kummer transformation (cf. Bailey [1] §3.2 and Page 98), to the following $3 F 2$-series
$F ( a , c , e ; b , d ) : = 3 F 2 1 + a , c , 1 2 + e 3 4 + b , 5 4 + d | 1 , Δ : = 1 2 + b + d − a − c − e > 0 σ : = b + d − a − c − e ≥ 0 ,$
where $a , b , c , d , e ∈ Z$ satisfying the conditions $a ≥ 0$ and $c > 0$ so that the both series involved are nonterminating. When $σ = b + d − a − c − e ≥ 0$, the series is convergent, because in this case the parameter excess $Δ = σ + 1 2 > 0$ (i.e., the sum of the denominator parameters minus that of the numerator ones).
Classically, there are three typical summation theorems (for the $3 F 2$-series) discovered by Dixon, Watson and Whipple (cf. Bailey [1] §3.1, §3.3 and §3.4). However, neither of them can evaluate the afore-displayed series in closed form. In particular, the formulae for the $3 F 2$-series presented in this paper are not present in the recent paper by the author [12], and two useful compendiums: ([13] §8.1.2 and [14] §7.4.4), where numerous closed formulae are collected for the $3 F 2 ( 1 )$ series with numerical parameters.
By applying the linearization method (cf. [15,16,17,18]), we shall transform, in the next section, the evaluation of $F$-series into the $Ω m , n$-series treated recently by the author [19]. The main results are summarized in the conclusive theorem as well as twenty closed formulae for the $F$-series. Finally in Section 3, analytic formulae for six further classes of exotic $3 F 2$-series will be provided by employing the Thomae and Kummer transformations (cf. Bailey [1] §3.2 and Page 98) to the $F$-series.
In order to ensure the accuracy, all the formulae appearing in this paper have been checked numerically by appropriately devised Mathematica commands.

## 2. Linearization Procedure for the $F$-Series

In this section, we shall reduce, by means of the linearization method (cf. [15,16,17,18]), the $F$-series to specific instances of a known $Ω m , n ( x , y )$ function, that has recently been examined by the author [19].

#### 2.1. $a = 0$

According to the Chu–Vandermonde convolution identity on binomial coefficients, it is routine to establish the following lemma.
Lemma 1
(Linear relation: $m ∈ N 0$).
$( A + n ) m = ∑ k = 0 m ( B + n ) k X k w h e r e X k = m k ( A − B ) m − k .$
Specifying the above relation to the equality
$( 1 + n ) a = ∑ k = 0 a ( c + n ) k X k ( a ) where X k ( a ) = a k ( 1 − c ) a − k$
and then substituting it into the $F$-series, we have the double series
$F ( a , c , e ; b , d ) = ∑ n = 0 ∞ 1 + a , c , 1 2 + e 1 , 3 4 + b , 5 4 + d n ∑ k = 0 a ( c + n ) k ( 1 + n ) a X k ( a ) = ∑ k = 0 a ( c ) k ( 1 ) a X k ( a ) ∑ n = 0 ∞ c + k , 1 2 + e 3 4 + b , 5 4 + d n .$
This results in the reduction formula as below.
Proposition 1
(Reduction formula from $a > 0$ to $a = 0$).
$F ( a , c , e ; b , d ) = ∑ k = 0 a ( − 1 ) a − c k c − 1 a − k F ( 0 , c + k , e ; b , d ) .$

#### 2.2. $b = d$

The $F$-series can further be reduced to the case $b = d$.
When $b > d$, we can specify Lemma 1 to the equality
$( 5 4 + d + n ) b − d = ∑ k = 0 b − d ( c + n ) k Y k ( b , d ) where Y k ( b , d ) = b − d k ( 5 4 − c + d ) b − d − k .$
Putting this inside the $F$-series, we have the double series
$F ( 0 , c , e ; b , d ) = ∑ n = 0 ∞ c , 1 2 + e 3 4 + b , 5 4 + d n ∑ k = 0 b − d ( c + n ) k ( 5 4 + d + n ) b − d Y k ( b , d ) = ∑ k = 0 b − d ( c ) k ( 5 4 + d ) b − d Y k ( b , d ) ∑ n = 0 ∞ c + k , 1 2 + e 3 4 + b , 5 4 + b n .$
This yields the following reduction formula.
Proposition 2
(Reduction formula from $b > d$ to $b = d$).
$F ( 0 , c , e ; b , d ) = ∑ k = 0 b − d b − d k ( c ) k ( 5 4 − c + d ) b − d − k ( 5 4 + d ) b − d F ( 0 , c + k , e ; b , b ) .$
Alternatively, for $b < d$, we can specify Lemma 1 to the equality
$( 3 4 + b + n ) d − b = ∑ k = 0 d − b ( c + n ) k Y k ( b , d ) where Y k ( b , d ) = d − b k ( 3 4 + b − c ) d − b − k .$
Substituting this into the $F$-series, we have the double series
$F ( 0 , c , e ; b , d ) = ∑ n = 0 ∞ c , 1 2 + e 3 4 + b , 5 4 + d n ∑ k = 0 d − b ( c + n ) k ( 3 4 + b + n ) d − b Y k ( b , d ) = ∑ k = 0 d − b ( c ) k ( 3 4 + b ) d − b Y k ( b , d ) ∑ n = 0 ∞ c + k , 1 2 + e 3 4 + d , 5 4 + d n .$
This gives rise to another reduction formula.
Proposition 3
(Reduction formula from $b < d$ to $b = d$).
$F ( 0 , c , e ; b , d ) = ∑ k = 0 d − b d − b k ( c ) k ( 3 4 + b − c ) d − b − k ( 3 4 + b ) d − b F ( 0 , c + k , e ; d , d ) .$

#### 2.3. $c = e$

The $F$-series can further be reduced to the case $c = e$. For this purpose, we have to show the following linearization lemma.
Lemma 2
(Linear relation: $m ∈ N 0$).
$( A + n ) m = ∑ k = 0 m B + 2 n k Z k w h e r e Z k = ∑ i = 0 k ( − 1 ) k − i k ! k i ( A − B − i 2 ) m .$
Proof.
By substitution, it suffices to evaluate the double sum
$S : = ∑ k = 0 m B + 2 n k ∑ i = 0 k ( − 1 ) k − i k ! k i ( A − B − i 2 ) m = ( A + n ) m .$
By exchanging the order of summations, we can reformulate it as
$S = ∑ i = 0 m B + 2 n i i ! ( A − B − i 2 ) m ∑ k = i m ( − 1 ) k − i B + 2 n − i k − i = ∑ i = 0 m ( − 1 ) m − i B + 2 n i i ! ( A − B − i 2 ) m B + 2 n − i − 1 m − i = B + 2 n m + 1 m ! ∑ i = 0 m ( − 1 ) m − i m i ( A − B − i 2 ) m B + 2 n − i = ( B + 2 n ) m + 1 m ! × m ! ( A + n ) m B + 2 n m + 1 = ( A + n ) m ,$
where the last line is justified by finite difference calculus (cf. [20,21]). □
First for $c < e$, we have from Lemma 2 the equality
$( 1 2 + c + n ) e − c = ∑ k = 0 e − c 2 b + 2 n + 1 2 k Z k ( b , c , e ) , where Z k ( b , c , e ) = ∑ i = 0 k ( − 1 ) k − i k ! k i ( c − b + 1 + 2 i 4 ) e − c .$
By inserting this into the $F$-series, we obtain the double series below
$F ( 0 , c , e ; b , b ) = ∑ n = 0 ∞ c , 1 2 + e 3 4 + b , 5 4 + b n ∑ k = 0 e − c 2 b + 2 n + 1 2 k ( 1 2 + c + n ) e − c Z k ( b , c , e ) = ∑ k = 0 e − c ( − 1 ) k ( − 1 2 − 2 b ) k ( 1 2 + c ) e − c Z k ( b , c , e ) ∑ n = 0 ∞ c , 1 2 + c 3 − 2 k 4 + b , 5 − 2 k 4 + b n .$
Writing the inner sum concerning n in terms of the $F$-series, we immediately establish the reduction formula as in the following proposition.
Proposition 4
(Reduction formula from $c < e$ to $c = e$).
$F ( 0 , c , e ; b , b ) = ∑ k = 0 e − c ( − 1 ) k ( − 1 2 − 2 b ) k ( 1 2 + c ) e − c Z k ( b , c , e ) F ( 0 , c , c ; b − k 2 , b − k 2 ) .$
When $c > e$ and $e > 0$, we infer from Lemma 2 that
$( e + n ) c − e = ∑ k = 0 c − e 2 b + 2 n + 1 2 k Z k ( b , c , e ) ,$
$where Z k ( b , c , e ) = ∑ i = 0 k ( − 1 ) k − i k ! k i ( e − b + 2 i − 1 4 ) c − e .$
Putting this inside the $F$-series, we can analogously treat the double series
$F ( 0 , c , e ; b , b ) = ∑ n = 0 ∞ c , 1 2 + e 3 4 + b , 5 4 + b n ∑ k = 0 c − e 2 b + 2 n + 1 2 k ( e + n ) c − e Z k ( b , c , e ) = ∑ k = 0 c − e ( − 1 ) k ( − 1 2 − 2 b ) k ( e ) c − e Z k ( b , c , e ) ∑ n = 0 ∞ e , 1 2 + e 3 − 2 k 4 + b , 5 − 2 k 4 + b n .$
Instead, for $c > e$ and $e ≤ 0$, reformulate first the $F$-series by reindexing
$F ( 0 , c , e ; b , b ) = F ( 0 , 1 + c − e , 1 ; 1 + b − e , 1 + b − e ) × c , 1 2 + e 3 4 + b , 5 4 + b 1 − e + ∑ n = 0 − e c , 1 2 + e 3 4 + b , 5 4 + b n .$
Then according to Lemma 2, we have another equality
$( 1 + n ) c − e = ∑ k = 0 c − e 5 2 + 2 b − 2 e + 2 n k Z k ( b , c , e ) ,$
where the connection coefficients $Z k ( b , c , e )$ coincide with those given by (1). Now, by substitution, we have another double series
$F ( 0 , 1 + c − e , 1 ; 1 + b − e , 1 + b − e ) = ∑ n = 0 ∞ 1 + c − e , 3 2 7 4 + b − e , 9 4 + b − e n ∑ k = 0 c − e 5 2 + 2 b − 2 e + 2 n k ( 1 + n ) c − e Z k ( b , c , e ) = ∑ k = 0 c − e ( − 1 ) k ( 2 e − 2 b − 5 2 ) k ( c − e ) ! Z k ( b , c , e ) ∑ n = 0 ∞ 1 , 3 2 7 − 2 k 4 + b − e , 9 − 2 k 4 + b − e n .$
Summing up, we have established the reduction formula to the case $c = e$.
Proposition 5
(Reduction formula from $c > e$ to $c = e$).
$e > 0 : F ( 0 , c , e ; b , b ) = ∑ k = 0 c − e ( − 1 ) k ( − 1 2 − 2 b ) k ( e ) c − e Z k ( b , c , e ) F ( 0 , e , e ; b − k 2 , b − k 2 ) , e ≤ 0 : F ( 0 , c , e ; b , b ) = ∑ n = 0 − e c , 1 2 + e 3 4 + b , 5 4 + b n + ∑ k = 0 c − e ( − 1 ) k ( 2 e − 2 b − 5 2 ) k ( c − e ) ! Z k ( b , c , e ) × c , 1 2 + e 3 4 + b , 5 4 + b 1 − e F ( 0 , 1 , 1 ; 1 + b − e − k 2 , 1 + b − e − k 2 ) .$
Observe that the parameter excess $Δ ≥ 1 2$ for the $F$-series is not diminished hitherto by the established reduction formulae. Consequently, all the $F$-series displayed on the right hand sides of Propositions 4 and 5 have the parameter excess $Δ ≥ 1 2$, and can be expressed as the following bisection series
$F ( 0 , c , c ; b , b ) = ∑ n = 0 ∞ ( 2 c ) 2 n ( 2 b + 3 2 ) 2 n = 1 2 × 2 F 1 1 , 2 c 3 2 + 2 b | 1 + 1 2 × 2 F 1 1 , 2 c 3 2 + 2 b | − 1 ,$
where $b , c ∈ N$ subject to the condition $b ≥ c$. Therefore, to evaluate the $F$-series explicitly, it suffices to do that for the above bisection series.

#### 2.4. $Ω m , n$-Series

In a recent paper [19], the author examined a more general series
$Ω m , n ( x , y ) : = 2 F 1 x , m − x n + 1 2 | y 2 where m , n ∈ Z$
and proved the following evaluation formula.
Theorem 1
(Chu [19] Theorems 4 and 8: Recurrence formula). For the two natural numbers m and n satisfying $m < n$, there holds the following formula
$Ω m , n ( x , y ) = ( 1 2 ) n y 2 n ∑ i = 0 n − m n − m i ( x ) i ( m − x ) n − m − i ( 2 x − n + i ) i ( m − 2 x − i ) n − m − i × ∑ k = 0 n ( − 1 ) n − k n k 2 x + 2 i − 2 k ( 2 x + 2 i − n − k ) n + 1 Ω 0 , 0 ( x + i − k , y ) ,$
where the series $Ω 0 , 0$ is evaluated by
$Ω 0 , 0 ( x , y ) = 2 F 1 x , − x 1 2 | y 2 = cos ( 2 x arcsin y ) .$
Hence, the $F$-series can be evaluated in terms of the $Ω$-series by the theorem below.
Theorem 2
($b ≥ c : b , c ∈ N$).
$F ( 0 , c , c ; b , b ) = 1 2 lim x → 1 Ω 2 c + 1 , 2 b + 1 ( x , 1 ) + 1 2 lim x → 1 Ω 2 c + 1 , 2 b + 1 ( x , − 1 ) w i t h Ω 0 , 0 ( x , 1 ) = cos ( π x ) a n d Ω 0 , 0 ( x , − 1 ) = cosh 2 x ln ( 1 + 2 ) .$

#### 2.5. Conclusive Theorem and Examples (Class-A)

Based on the preceding reduction formulae, we may evaluate, for any quintuple integers $a , b , c , d , e ∈ Z$ subject to $a ≥ 0$, $c > 0$ and $σ = b + d − a − c − e > 0$, the $F$-series by carrying out the following procedure:
• Step-A: If $a = 0$, go directly to Step-B. Otherwise for $a > 0$, according to Proposition 1, express $F ( a , c , e ; b , d )$ in terms of $F ( 0 , c , e ; b , d )$, and then go to Step-B.
• Step-B: By means of Propositions 2 and 3, express $F ( 0 , c , e ; b , d )$ in terms of $F ( 0 , c , e ; b , b )$, and then go to Step-C.
• Step-C: In virtu of Propositions 4 and 5, express $F ( 0 , c , e ; b , b )$ in terms of $F ( 0 , c , c ; b , b )$, and then go to Step-D.
• Step-D: Finally by applying Theorems 1 and 2, evaluate $F ( 0 , c , c ; b , b )$ explicitly in terms of the $Ω$-series.
Therefore, we have validated the conclusive theorem as below.
Theorem 3
(Conclusion). For any quintuple integers
$a , b , c , d , e ∈ Z s u b j e c t t o a ≥ 0 , c > 0 a n d σ = b + d − a − c − e > 0 ,$
the nonterminating $F ( a , c , e ; b , d )$ series can always be evaluated by finitely linear sums of trigonometric function $cos ( π x )$ and hyperbolic function $cosh 2 x ln ( 1 + 2 )$, where $x ∈ Z$ and the coefficients are rational numbers.
According to the afore-described procedure, we have written appropriate Mathematica commands to determine explicitly closed form expressions for $F ( a , c , e ; b , d )$ series. Twenty summation formulae are displayed below, where the argument “1” will be suppressed from the notation of $3 F 2$-series for the sake of brevity. We shall call these series “Class-A”. Among them, an equivalent form of A5 has been obtained by Campbell and Abrarov ([11] Equation (18)).
$A 1 . 3 F 2 [ 1 , 1 , 1 2 ; 5 4 , 7 4 ] = 3 2 log ( 1 + 2 ) . A 2 . 3 F 2 [ 1 , 1 , 1 2 ; 7 4 , 9 4 ] = − 5 1 − 2 log ( 1 + 2 ) . A 3 . 3 F 2 [ 1 , 1 , 1 2 ; 5 4 , 11 4 ] = − 7 15 1 − 3 2 log ( 1 + 2 ) . A 4 . 3 F 2 [ 1 , 1 , 3 2 ; 5 4 , 11 4 ] = 7 12 2 + 3 2 log ( 1 + 2 ) . A 5 . 3 F 2 [ 1 , 1 , 3 2 ; 7 4 , 9 4 ] = 15 4 2 − 2 log ( 1 + 2 ) . A 6 . 3 F 2 [ 1 , 1 , 3 2 ; 9 4 , 11 4 ] = 35 6 4 − 3 2 log ( 1 + 2 ) . A 7 . 3 F 2 [ 1 , 1 , − 1 2 ; 3 4 , 5 4 ] = 1 3 1 − 2 log ( 1 + 2 ) . A 8 . 3 F 2 [ 1 , 1 , − 1 2 ; 1 4 , 7 4 ] = 3 5 1 − 3 2 log ( 1 + 2 ) . A 9 . 3 F 2 [ 1 , 1 , − 1 2 ; 7 4 , 9 4 ] = − 3 7 3 − 4 2 log ( 1 + 2 ) . A 10 . 3 F 2 [ 1 , 1 , − 3 2 ; 3 4 , 5 4 ] = 1 35 3 − 4 2 log ( 1 + 2 ) .$
$A 11 . 3 F 2 [ 1 , 2 , 1 2 ; 7 4 , 9 4 ] = 5 8 2 + 2 log ( 1 + 2 ) . A 12 . 3 F 2 [ 1 , 2 , − 1 2 ; 5 4 , 7 4 ] = 3 20 2 − 2 log ( 1 + 2 ) . A 13 . 3 F 2 [ 1 , 2 , − 1 2 ; 3 4 , 9 4 ] = 5 84 2 − 5 2 log ( 1 + 2 ) . A 14 . 3 F 2 [ 1 , 2 , − 1 2 ; 7 4 , 9 4 ] = 1 14 8 + 2 log ( 1 + 2 ) . A 15 . 3 F 2 [ 1 , 2 , − 3 2 ; 3 4 , 9 4 ] = − 5 77 1 + 2 log ( 1 + 2 ) . A 16 . 3 F 2 [ 2 , 2 , 1 2 ; 7 4 , 13 4 ] = 135 224 6 − 2 log ( 1 + 2 ) . A 17 . 3 F 2 [ 2 , 2 , − 1 2 ; 5 4 , 11 4 ] = 7 48 2 − 3 2 log ( 1 + 2 ) . A 18 . 3 F 2 [ 2 , 2 , − 1 2 ; 9 4 , 11 4 ] = 1 24 2 + 9 2 log ( 1 + 2 ) . A 19 . 3 F 2 [ 2 , 2 , − 3 2 ; 7 4 , 13 4 ] = 1 22 8 − 3 2 log ( 1 + 2 ) . A 20 . 3 F 2 [ 2 , 2 , − 3 2 ; 11 4 , 13 4 ] = − 1 13 13 − 15 2 log ( 1 + 2 ) .$

## 3. The Thomae and Kummer Transformations

In the classical theory of hypergeometric series, the Thomae and Kummer transformations are fundamental (cf. Bailey [1] §3.2 and Page 98, where $σ = b + d − a − c − e$):
$3 F 2 a , c , e b , d | 1 = 3 F 2 σ , b − a , d − a c + σ , e + σ | 1 Γ σ , b , d a , c + σ , e + σ$
$3 F 2 a , c , e b , d | 1 = 3 F 2 a , b − c , b − e σ + a , b | 1 Γ σ , d σ + a , d − a .$
They will be applied to the $F$-series to evaluete six classes of exotic $3 F 2$-series.

#### 3.1. Class B

Applying the Kummer transformation (4), we can express the following “Class-B” series in terms of the $F$-series (where $σ = b + d − a − c − e$):
$3 F 2 1 + a , c + 1 4 , e + 3 4 b + 3 2 , d + 5 4 | 1 = Γ b + 3 2 , σ + 3 4 b − a + 1 2 , σ + a + 7 4 × 3 F 2 1 + a , d − c + 1 , d − e + 1 2 d + 5 4 , σ + a + 7 4 | 1 .$
Then we can derive the following closed formulae for these series (except for divergent series) from those displayed in “Class A”.
$B 1 . 3 F 2 [ 1 , 1 4 , 3 4 ; 3 2 , 5 4 ] = 2 log ( 1 + 2 ) . B 2 . 3 F 2 [ 1 , 1 4 , 7 4 ; 5 2 , 5 4 ] = 2 5 1 + 2 2 log ( 1 + 2 ) . B 3 . 3 F 2 [ 1 , 1 4 , 7 4 ; 5 2 , 9 4 ] = 3 2 2 − 2 log ( 1 + 2 ) . B 4 . 3 F 2 [ 1 , 3 4 , 5 4 ; 3 2 , 9 4 ] = 5 2 2 − 2 log ( 1 + 2 ) . B 5 . 3 F 2 [ 1 , 3 4 , 5 4 ; 5 2 , 9 4 ] = 5 4 − 3 2 log ( 1 + 2 ) . B 6 . 3 F 2 [ 1 , 7 4 , 9 4 ; 5 2 , 13 4 ] = 9 5 8 − 5 2 log ( 1 + 2 ) . B 7 . 3 F 2 [ 2 , 1 4 , 7 4 ; 7 2 , 5 4 ] = 2 9 4 + 3 2 log ( 1 + 2 ) . B 8 . 3 F 2 [ 2 , 3 4 , 5 4 ; 5 2 , 9 4 ] = 5 4 − 2 + 3 2 log ( 1 + 2 ) . B 9 . 3 F 2 [ 2 , 5 4 , 7 4 ; 7 2 , 9 4 ] = 5 2 − 2 log ( 1 + 2 ) . B 10 . 3 F 2 [ 2 , 9 4 , 11 4 ; 9 2 , 13 4 ] = 30 4 − 3 2 log ( 1 + 2 ) .$

#### 3.2. Class C

By means of the Kummer transformation (4), we can express the “Class-C” series below in terms of the $F$-series (where $σ = b + d − a − c − e$):
$3 F 2 1 + a , c + 1 4 , e + 3 4 b + 3 2 , d + 3 4 | 1 = Γ σ + 1 4 , b + 3 2 σ + a + 5 4 , b − a + 1 2 × 3 F 2 1 + a , d − e , d − c + 1 2 d + 3 4 , σ + a + 5 4 | 1 .$
Then the closed formulae below for these series ( except for divergent series) follow directly from those recorded in “Class A”.
$C 1 . 3 F 2 [ 1 , 1 4 , 3 4 ; 3 2 , 7 4 ] = 3 2 2 − 2 log ( 1 + 2 ) . C 2 . 3 F 2 [ 1 , 1 4 , 3 4 ; 5 2 , 7 4 ] = 3 5 8 − 5 2 log ( 1 + 2 ) . C 3 . 3 F 2 [ 1 , 3 4 , 5 4 ; 3 2 , 7 4 ] = 3 2 log ( 1 + 2 ) . C 4 . 3 F 2 [ 1 , 3 4 , 9 4 ; 5 2 , 7 4 ] = 6 5 1 + 2 2 log ( 1 + 2 ) . C 5 . 3 F 2 [ 1 , 5 4 , − 1 4 ; 3 2 , 3 4 ] = 2 3 1 − 2 log ( 1 + 2 ) . C 6 . 3 F 2 [ 1 , 5 4 , − 1 4 ; 3 2 , 7 4 ] = 1 4 2 + 2 log ( 1 + 2 ) . C 7 . 3 F 2 [ 1 , 5 4 , − 1 4 ; 5 2 , 3 4 ] = 2 7 5 − 2 2 log ( 1 + 2 ) . C 8 . 3 F 2 [ 2 , 3 4 , 5 4 ; 5 2 , 7 4 ] = 3 2 2 + 2 log ( 1 + 2 ) . C 9 . 3 F 2 [ 2 , 3 4 , 9 4 ; 7 2 , 7 4 ] = 6 7 8 + 2 log ( 1 + 2 ) . C 10 . 3 F 2 [ 2 , 3 4 , 13 4 ; 7 2 , 11 4 ] = 1 2 2 + 9 2 log ( 1 + 2 ) .$

#### 3.3. Class D

By virtue of the Thomae transformation (3), we can express the following “Class-D” series in terms of the $F$-series (where $σ = b + d − a − c − e$):
$3 F 2 a + 1 2 , c + 1 4 , e + 3 4 b + 1 , d + 1 2 | 1 = Γ σ , b + 1 , d + 1 2 a + 1 2 , σ + c + 1 4 σ + e + 3 4 × 3 F 2 σ , d − a , b − a + 1 2 σ + c + 1 4 , σ + e + 3 4 | 1 .$
Then we find the closed formulae below for these series ( except for divergent series) as consequences of those produced in “Class A”.
$D 1 . 3 F 2 [ 1 4 , 3 4 , 1 2 ; 1 , 3 2 ] = 4 log ( 1 + 2 ) π . D 2 . 3 F 2 [ 1 4 , 7 4 , 1 2 ; 2 , 3 2 ] = 8 2 + 3 log ( 1 + 2 ) 9 π . D 3 . 3 F 2 [ 1 4 , 7 4 , 1 2 ; 1 , 5 2 ] = 2 2 + 9 log ( 1 + 2 ) 5 π . D 4 . 3 F 2 [ 1 4 , 7 4 , 3 2 ; 1 , 7 2 ] = 8 2 2 + 3 log ( 1 + 2 ) 9 π . D 5 . 3 F 2 [ 3 4 , 5 4 , 1 2 ; 1 , 5 2 ] = 2 2 + log ( 1 + 2 ) π .$
$D 6 . 3 F 2 [ 3 4 , 5 4 , 1 2 ; 2 , 3 2 ] = 8 2 − log ( 1 + 2 ) π . D 7 . 3 F 2 [ 3 4 , 5 4 , 3 2 ; 1 , 7 2 ] = 8 4 2 + log ( 1 + 2 ) 7 π . D 8 . 3 F 2 [ 5 4 , − 1 4 , 1 2 ; 1 , 3 2 ] = 4 2 2 − log ( 1 + 2 ) 3 π . D 9 . 3 F 2 [ 5 4 , − 1 4 , 3 2 ; 1 , 5 2 ] = 4 5 2 − 4 log ( 1 + 2 ) 7 π . D 10 . 3 F 2 [ 5 4 , 7 4 , 3 2 ; 2 , 7 2 ] = 16 2 − log ( 1 + 2 ) π .$
Observing that the parameter excess of the $3 F 2$-series displayed on the right hand side of (5) equals $Δ = 1 2 + a$, the equality (5) valid only when $a ≥ 0$ and $σ ≥ 0$. It remains a problem to evaluate, for $a < 0$, the $3 F 2$-series on the left of (5). This can also be resolved by the linearization method.
According to the Pfaff–Saalschütz summation theorem (cf. Bailey [1] §2.2), it is not hard to confirm the linear relation in the following lemma.
Lemma 3
(Linear relation: $m ∈ N 0$).
$( A + n ) m = ∑ k = 0 m n k ( B + n ) m − k X k , w h e r e X k = ( − 1 ) k m k ( A ) m ( A − B ) k ( B ) m ( A ) k .$
By specializing this to the equality
$( 1 + b + n ) − a = ∑ k = 0 − a n k ( 1 2 + a + n ) − a − k X a k , where X k ( a ) = ( − a ) ! − 1 2 − a a − b − 1 2 k b − a − a − k$
and then substituting it into the $3 F 2$-series, we may manipulate the double sum
$3 F 2 1 2 + a , 1 4 + c , 3 4 + e 1 + b , 1 2 + d | 1 = ∑ n = 0 ∞ 1 2 + a , 1 4 + c , 3 4 + e 1 , 1 + b , 1 2 + d n ∑ k = 0 − a n k ( 1 2 + a + n ) − a − k ( 1 + b + n ) − a X k ( a ) = ∑ k = 0 − a ( 1 2 + a ) − a − k ( 1 + b ) − a X k ( a ) ∑ n = 0 ∞ n k n ! 1 2 − k , 1 4 + c , 3 4 + e 1 − a + b , 1 2 + d n .$
Performing the replacement $n → n + k$, we can express the last sum with respect to n as
$1 2 − k , 1 4 + c , 3 4 + e 1 − a + b , 1 2 + d k 3 F 2 1 2 , 1 4 + c + k , 3 4 + e + k 1 − a + b + k , 1 2 + d + k | 1 .$
Therefore, we have established, after some simplifications, the following transformation formula.
Theorem 4
(Reduction formula from $a < 0$ to $a = 0$).
$3 F 2 1 2 + a , 1 4 + c , 3 4 + e 1 + b , 1 2 + d | 1 = ∑ k = 0 − a a , 1 2 − a + b , 1 4 + c , 3 4 + e 1 , 1 − a + b , 1 + b , 1 2 + d k × 3 F 2 1 2 , 1 4 + c + k , 3 4 + e + k 1 − a + b + k , 1 2 + d + k | 1 .$
It should be emphasized that under this transformation, the parameter excess $Δ = σ = b + d − a − c − e$ remains invariant for all the $3 F 2$-series. However the $3 F 2$-series on the right belongs to Class-D and can therefore be evaluated by (5). Ten more formulae are recorded below.
$D 11 . 3 F 2 [ − 1 2 , 1 4 , − 1 4 ; 1 , 3 2 ] = 13 2 + log ( 1 + 2 ) 6 π . D 12 . 3 F 2 [ − 1 2 , 1 4 , 3 4 ; 1 , 1 2 ] = 2 + log ( 1 + 2 ) π . D 13 . 3 F 2 [ − 1 2 , 1 4 , 3 4 ; 1 , 3 2 ] = 2 + 5 log ( 1 + 2 ) 2 π . D 14 . 3 F 2 [ − 1 2 , 1 4 , 7 4 ; 1 , 3 2 ] = 5 2 + 9 log ( 1 + 2 ) 6 π . D 15 . 3 F 2 [ − 1 2 , 3 4 , 5 4 ; 1 , 3 2 ] = 3 2 − log ( 1 + 2 ) 2 π . D 16 . 3 F 2 [ − 1 2 , 3 4 , − 3 4 ; 1 , 1 2 ] = 5 2 + 9 log ( 1 + 2 ) 3 π . D 17 . 3 F 2 [ − 1 2 , 3 4 , − 3 4 ; 2 , 1 2 ] = 34 2 + 42 log ( 1 + 2 ) 21 π . D 18 . 3 F 2 [ − 1 2 , 5 4 , 7 4 ; 2 , 3 2 ] = 7 2 + 3 ln ( 1 + 2 ) 9 π . D 19 . 3 F 2 [ − 1 2 , − 3 4 , 7 4 ; 1 3 2 ] = 43 2 + 87 log ( 1 + 2 ) 30 π . D 20 . 3 F 2 [ − 3 2 , − 1 4 , − 3 4 ; 1 , 1 2 ] = 31 2 − 37 log ( 1 + 2 ) 8 π .$
Campbell, D’Aurizio and Sondow [9,10,22] discovered some formulae in Class-D.
• The formula D1 has been found by them in ([9] Equation (10)), where they also conjectured D12. For this last evaluation, five different proofs have been provided by the same authors [10].
• By making use of beta integrals, Campbell recoded in ([22] Theorems 2,3,7 and Example 12) four formulae. The first one ([22] Theorem 2) is corrected by D18. The second one ([22] Theorem 3) is incorrect. The third one ([22] Theorem 7) is simplified by D2. The fourth one ([22] Example 12) is too complicated to reproduce here.

#### 3.4. Class E

Again in view of the Thomae transformation (3), we can express the “Class-E” series below in terms of the $F$-series (where $σ = b + d − a − c − e$):
$3 F 2 a + 1 2 , c + 1 4 , e + 3 4 b + 1 2 , d + 3 2 | 1 = Γ σ + 1 2 , b + 1 2 , d + 3 2 a + 1 2 , σ + c + 3 4 , σ + e + 5 4 × 3 F 2 1 + d − a , b − a , σ + 1 2 σ + c + 3 4 , σ + e + 5 4 | 1 .$
Consequently, the closed formulae below for these series ( except for divergent series) can be deduced from those exhibited in “Class A”. Among them, E2 simplifies a formula of Campbell ([22] Example 5).
$E 1 . 3 F 2 [ 1 4 , 3 4 , 1 2 ; 3 2 , 3 2 ] = 2 2 − 2 log ( 1 + 2 ) . E 2 . 3 F 2 [ 1 4 , 7 4 , 1 2 ; 3 2 , 3 2 ] = 1 3 4 2 − 2 log ( 1 + 2 ) . E 3 . 3 F 2 [ 1 4 , 7 4 , 3 2 ; 5 2 , 5 2 ] = 12 25 8 2 − 10 log ( 1 + 2 ) . E 4 . 3 F 2 [ 3 4 , 5 4 , 1 2 ; 3 2 , 3 2 ] = 2 log ( 1 + 2 ) . E 5 . 3 F 2 [ 3 4 , 5 4 , 3 2 ; 5 2 , 5 2 ] = 4 4 2 − 6 log ( 1 + 2 ) . E 6 . 3 F 2 [ 3 4 , 9 4 , 1 2 ; 3 2 , 5 2 ] = 1 5 2 + 9 log ( 1 + 2 ) . E 7 . 3 F 2 [ 5 4 , 7 4 , 1 2 ; 3 2 , 5 2 ] = 2 + log ( 1 + 2 ) . E 8 . 3 F 2 [ 5 4 , 11 4 , 1 2 ; 5 2 , 5 2 ] = 9 14 3 2 − log ( 1 + 2 ) . E 9 . 3 F 2 [ 7 4 , 9 4 , 3 2 ; 5 2 , 7 2 ] = 12 2 − log ( 1 + 2 ) . E 10 . 3 F 2 [ 9 4 , 11 4 , 5 2 ; 7 2 , 9 2 ] = 40 4 2 − 6 log ( 1 + 2 ) .$
Analogous to the series in Class-D, the parameter excess of the $3 F 2$-series displayed on the right hand side of (6) equals $Δ = 1 2 + a$, which converges only when $a ≥ 0$. We can also evaluate that $3 F 2$-series by reducing the case $a < 0$ to $a = 0$.
By means of Lemma 3, we have the equality
$( 1 2 + b + n ) − a = ∑ k = 0 − a n k ( 1 2 + a + n ) − a − k X a k , where X k ( a ) = ( − a ) ! − 1 2 − a a − b k b − a − 1 2 − a − k$
and then insert it in the $3 F 2$-series, we can handle the double sum
$3 F 2 1 2 + a , 1 4 + c , 3 4 + e 1 2 + b , 3 2 + d | 1 = ∑ n = 0 ∞ 1 2 + a , 1 4 + c , 3 4 + e 1 , 1 2 + b , 3 2 + d n ∑ k = 0 − a n k ( 1 2 + a + n ) − a − k ( 1 2 + b + n ) − a X k ( a ) = ∑ k = 0 − a ( 1 2 + a ) − a − k ( 1 2 + b ) − a X k ( a ) ∑ n = 0 ∞ n k n ! 1 2 − k , 1 4 + c , 3 4 + e 1 2 − a + b , 3 2 + d n .$
Making the replacement $n → n + k$, we can express the last sum as
$1 2 − k , 1 4 + c , 3 4 + e 1 2 − a + b , 3 2 + d k 3 F 2 1 2 , 1 4 + c + k , 3 4 + e + k 1 2 − a + b + k , 3 2 + d + k | 1 .$
After some simplifications, we establish the transformation below.
Theorem 5
(Reduction formula from $a < 0$ to $a = 0$).
$3 F 2 1 2 + a , 1 4 + c , 3 4 + e 1 2 + b , 3 2 + d | 1 = ∑ k = 0 − a a , b − a , 1 4 + c , 3 4 + e 1 , 1 2 − a + b , 1 2 + b , 3 2 + d k × 3 F 2 1 2 , 1 4 + c + k , 3 4 + e + k 1 2 − a + b + k , 3 2 + d + k | 1 .$
Under this transformation, the parameter excess $Δ = σ = b + d − a − c − e$ remains invariant for all the $3 F 2$-series involved. However the $3 F 2$-series on the right belongs to Class-E and can therefore be evaluated by (6). We record ten more examples.
$E 11 . 3 F 2 [ − 1 2 , 1 4 , 3 4 ; 1 2 , 3 2 ] = 6 − 2 log ( 1 + 2 ) 4 2 . E 12 . 3 F 2 [ − 1 2 , 1 4 , 7 4 ; 1 2 , 3 2 ] = 14 − 5 2 log ( 1 + 2 ) 12 2 . E 13 . 3 F 2 [ − 1 2 , 3 4 , 5 4 ; 1 2 , 3 2 ] = 2 − 3 2 log ( 1 + 2 ) 4 2 . E 14 . 3 F 2 [ − 1 2 , 3 4 , 5 4 ; 3 2 , 3 2 ] = 2 + 5 2 log ( 1 + 2 ) 8 2 . E 15 . 3 F 2 [ − 1 2 , 5 4 , 7 4 ; 1 2 , 5 2 ] = 3 2 + 5 2 log ( 1 + 2 ) − 16 2 . E 16 . 3 F 2 [ − 1 2 , 5 4 , 7 4 ; 3 2 , 3 2 ] = 10 − 7 2 log ( 1 + 2 ) 24 2 . E 17 . 3 F 2 [ − 3 2 , 3 4 , 5 4 ; 1 2 , 3 2 ] = 10 − 39 2 log ( 1 + 2 ) 128 2 . E 18 . 3 F 2 [ − 3 2 , 3 4 , 5 4 ; 1 2 , 5 2 ] = 3 62 − 37 2 log ( 1 + 2 ) 256 2 . E 19 . 3 F 2 [ − 3 2 , 5 4 , 7 4 ; 3 2 , 3 2 ] = 62 − 37 2 log ( 1 + 2 ) 512 2 . E 20 . 3 F 2 [ − 3 2 , 5 4 , − 1 4 ; 1 2 , 3 2 ] = 3 42 + 41 2 log ( 1 + 2 ) 128 2 .$

#### 3.5. Class F

By invoking the Kummer transformation (4), we can express the “Class-F” series below in terms of the $F$-series (where $σ = b + d − a − c − e$):
$3 F 2 a + 1 2 , c + 3 4 , e + 3 4 b + 1 , d + 7 4 | 1 = Γ b + 1 , σ + 3 4 b − a + 1 2 , σ + a + 5 4 × 3 F 2 1 + d − c , 1 + d − e , a + 1 2 σ + a + 5 4 , d + 7 4 | 1 .$
Then the closed formulae below for these series ( except for divergent series) can be established from those shown in “Class A”. Among them, the formula F10 is due to Campbell and Abrarov ([11] Corollary 5).
$F 1 . 3 F 2 [ − 1 2 , − 1 4 , − 1 4 ; 1 , 3 4 ] = 2 π 5 2 − 4 log ( 1 + 2 ) Γ ( 1 4 ) 2 . F 2 . 3 F 2 [ − 1 2 , 3 4 , − 1 4 ; 1 , 7 4 ] = 3 π 8 2 + 2 log ( 1 + 2 ) 5 Γ ( 1 4 ) 2 . F 3 . 3 F 2 [ − 1 2 , 3 4 , 3 4 ; 1 , 7 4 ] = 6 π 2 + 4 log ( 1 + 2 ) 5 Γ ( 1 4 ) 2 . F 4 . 3 F 2 [ − 1 2 , 3 4 , 7 4 ; 1 , 11 4 ] = 7 π 4 2 + 6 log ( 1 + 2 ) 15 Γ ( 1 4 ) 2 . F 5 . 3 F 2 [ 1 2 , − 1 4 , − 1 4 ; 1 , 3 4 ] = 2 π 4 2 − 2 log ( 1 + 2 ) Γ ( 1 4 ) 2 . F 6 . 3 F 2 [ 1 2 , − 1 4 , − 1 4 ; 1 , 7 4 ] = 9 π 3 2 − log ( 1 + 2 ) 4 Γ ( 1 4 ) 2 . F 7 . 3 F 2 [ 1 2 , 3 4 , − 1 4 ; 1 , 7 4 ] = 3 π 2 + log ( 1 + 2 ) Γ ( 1 4 ) 2 . F 8 . 3 F 2 [ 1 2 , 3 4 , 3 4 ; 1 , 7 4 ] = 12 π log ( 1 + 2 ) Γ ( 1 4 ) 2 . F 9 . 3 F 2 [ 1 2 , 3 4 , 7 4 ; 1 , 11 4 ] = 7 π 2 + 9 log ( 1 + 2 ) 5 Γ ( 1 4 ) 2 . F 10 3 F 2 [ 3 2 , 3 4 , − 1 4 ; 1 , 7 4 ] = 3 π 3 2 − log ( 1 + 2 ) 2 Γ ( 1 4 ) 2 .$

#### 3.6. Class G

Finally, by employing the Kummer transformation (4), we can express the “Class-G” series below in terms of the $F$-series (where $σ = b + d − a − c − e$):
$3 F 2 a + 1 2 , c + 1 4 , e + 1 4 b + 1 , d + 1 4 | 1 = Γ b + 1 , σ + 1 4 b − a + 1 2 , σ + a + 3 4 × 3 F 2 d − c , d − e , a + 1 2 d + 1 4 , σ + a + 3 4 | 1 .$
Then the closed formulae below for these series ( except for divergent series) can be shown from those displayed in “Class A”. Among them, the formula G8 is due to Campbell and Abrarov ([11] Corollary 4), who evaluated also another similar series ([11] Corollary 6).
$G 1 . 3 F 2 [ − 3 2 , 5 4 , 5 4 ; 1 , 13 4 ] = 5 π 4 2 − 3 log ( 1 + 2 ) 44 Γ ( 3 4 ) 2 . G 2 . 3 F 2 [ − 1 2 , 1 4 , − 3 4 ; 1 , 5 4 ] = π 2 2 + 3 log ( 1 + 2 ) 6 Γ ( 3 4 ) 2 . G 3 . 3 F 2 [ 1 2 , 1 4 , − 3 4 ; 1 , 5 4 ] = π 2 + 9 log ( 1 + 2 ) 12 Γ ( 3 4 ) 2 . G 4 . 3 F 2 [ 1 2 , 1 4 , 1 4 ; 1 , 5 4 ] = π log ( 1 + 2 ) Γ ( 3 4 ) 2 . G 5 . 3 F 2 [ 1 2 , 1 4 , 1 4 ; 2 , 5 4 ] = 2 π − 2 + 6 log ( 1 + 2 ) 9 Γ ( 3 4 ) 2 .$
$G 6 . 3 F 2 [ 1 2 , 1 4 , 5 4 ; 2 , 9 4 ] = 5 π 2 − log ( 1 + 2 ) 3 Γ ( 3 4 ) 2 . G 7 . 3 F 2 [ 1 2 , 5 4 , 9 4 ; 2 , 13 4 ] = 3 π − 2 + 5 log ( 1 + 2 ) 7 Γ ( 3 4 ) 2 . G 8 . 3 F 2 [ 3 2 , 1 4 , 5 4 ; 1 , 9 4 ] = 5 π 3 2 − log ( 1 + 2 ) 8 Γ ( 3 4 ) 2 G 9 . 3 F 2 [ 3 2 , 5 4 , 5 4 ; 2 , 9 4 ] = 5 π 2 2 − 2 log ( 1 + 2 ) Γ ( 3 4 ) 2 . G 10 . 3 F 2 [ 3 2 , 5 4 , 9 4 ; 3 , 13 4 ] = 6 π − 6 2 + 10 log ( 1 + 2 ) Γ ( 3 4 ) 2 .$

By combining the linearization method with the Kummer and Thomae transformations, we present 100 explicit formulae for 7 classes of nonterminating $3 F 2 ( 1 )$-series. They may potentially find applications in mathematics and physics as other mathematical formulae. Further explorations are encouraged to enrich this bank database of hypergeometric series identities.

## Funding

This research received no external funding.

Not applicable.

## Conflicts of Interest

The author declares no conflict of interest.

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