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Article

Further Closed Formulae of Exotic 3F2-Series

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

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 ( 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 .

Concluding Comments

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.

Data Availability Statement

Not applicable.

Conflicts of Interest

The author declares no conflict of interest.

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