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Theory and Modern Applications

Another class of nonterminating \(_{3}F_{2}\)-series with a free argument

Abstract

By means of the linearization method, we evaluate another class of nonterminating \(_{3}F_{2}\)-series with a free argument x and two perturbing integer parameters m and n.

1 Introduction and motivation

There has always been a strong interest in discovering novel summation formulae for (generalized) hypergeometric series due to their broad variety of applications in mathematics, physics, and computer science (see [57, 13, 14, 1921, 23]). The purpose of this paper is to evaluate, in closed forms, the following class of nonterminating \(_{3}F_{2}\)-series with a free variable x (with \(|x|<1\) for convergence) and two perturbing integer parameters m and n:

Ω m , n (a,x): = 3 F 2 [ a , a + 1 3 , a 1 3 1 2 + m , 3 a + n | x 2 ] ,
(1)

where, according to Bailey [2, §2.1], the classical hypergeometric series reads as

F p 1 + p [ a 0 , a 1 , , a p b 1 , , b p | z ] = k = 0 ( a 0 ) k ( a 1 ) k ( a p ) k k ! ( b 1 ) k ( b p ) k z k .

Denote by \(\mathbb{Z}\) and \(\mathbb{N}\), respectively, sets of integers and natural numbers with \(\mathbb{N}_{0}=\{0\}\cup \mathbb{N}\). For indeterminate y and \(n\in \mathbb{Z}\), the rising and falling factorials are defined by the following quotients of Euler’s Γ-function:

$$\begin{aligned} (x)_{n}=\frac{\Gamma (x+n)}{\Gamma (x)} \quad\text{and}\quad \langle {x} \rangle _{n}=\frac{\Gamma (1+x)}{\Gamma (1+x-n)}, \end{aligned}$$

where the multiparameter notation for the former one will be abbreviated to

[ A , B , , C α , β , , γ ] n = ( A ) n ( B ) n ( C ) n ( α ) n ( β ) n ( γ ) n .

Our work is motivated by Lambert’s binomial series (see Riordan [22, §4.5] and [1, 810, 15, 20]) which is well known in classical analysis. Let u and v be the two variables related through the equation \(u=v/(1+v)^{\beta }\). Then

$$\begin{aligned} &\phi _{\alpha }(u):=(1+v)^{\alpha }=\sum_{k=0}^{\infty } \frac{\alpha }{\alpha +k\beta } \binom{\alpha +k\beta }{k}u^{k}, \\ &\psi _{\alpha }(u):=\frac{(1+v)^{\alpha +1}}{1+v-\beta v} =\sum_{k=0}^{ \infty } \binom{\alpha +k\beta }{k}u^{k}. \end{aligned}$$

By the bisection of series, we have further four generating functions

$$\begin{aligned} &\sum_{k=0}^{\infty } \frac{\alpha }{\alpha +2\beta k} \binom{\alpha +2\beta k}{2k}u^{2k} = \frac{\phi _{\alpha }(u)+\phi _{\alpha }(-u)}{2}, \\ &\sum_{k=0}^{\infty } \binom{\alpha +2\beta k}{2k}u^{2k} = \frac{\psi _{\alpha }(u)+\psi _{\alpha }(-u)}{2}; \\ &\sum_{k=0}^{\infty } \frac{\alpha }{\alpha +\beta (2k+1)} \binom{\alpha +\beta (2k+1)}{2k+1}u^{2k+1} = \frac{\phi _{\alpha }(u)-\phi _{\alpha }(-u)}{2}, \\ &\sum_{k=0}^{\infty } \binom{\alpha +\beta (2k+1)}{2k+1}u^{2k+1} = \frac{\psi _{\alpha }(u)-\psi _{\alpha }(-u)}{2}. \end{aligned}$$

Specifying with \(\beta =\frac{3}{2}\), making the replacements \(u\to \frac{2x}{3\sqrt{3}}, v\to y\), and then letting

$$\begin{aligned} \alpha \to 3a-1, \alpha \to 3a-2, \alpha \to 3a-\frac{5}{2}, \alpha \to 3a- \frac{7}{2}, \end{aligned}$$

respectively, in the above four equations, we get four hypergeometric formulae:

figure a

Here and forth, x and y are two variables related via equations

$$\begin{aligned} \pm {x}=\frac{3\sqrt{3}y_{\pm }}{2\sqrt{(1+y_{\pm })^{3}}}, \end{aligned}$$
(2)

where \(y_{\pm }\) are computed from x through the fundamental algebraic relationship

$$\begin{aligned} \frac{2x}{3\sqrt{3}}=\frac{y}{(1+y)^{3/2}} \quad\text{or equivalently}\quad \biggl( \frac{2x}{y} \biggr)^{2}= \biggl( \frac{3}{1+y} \biggr)^{3}. \end{aligned}$$

Recall that the hypergeometric \(_{3}F_{2}(x^{2})\)-series converge (generically) only if their argument is less than 1 in magnitude. Therefore x is restricted to \((-1,1)\). There are exactly two solutions \(y_{+}\) and \(y_{-}\) of the above equation in the region \((-1/4,2)\) whenever x satisfies \(-1< x<1\). By equating both members of the last equation to \(t^{6}\), we can parameterize the algebraic “xy curve” by rational functions:

$$\begin{aligned} x=\frac{t}{2}\bigl(3-t^{2}\bigr)\quad \text{and}\quad y= \frac{3-t^{2}}{t^{2}}. \end{aligned}$$

The portions of the curve with \(t\in (-2,-1)\) and \(t\in (1,2)\) lie, in the “xy plane”, in the abovementioned region. For any x, the corresponding \(y_{\pm }\) are the y-coordinates of the points \((x,y)\) that lie on these two branches that are illustrated in the Fig. 1.

Figure 1
figure 1

The “\(x-y\)” curve

The four identities of \(_{3}F_{2}\)-series highlighted in the last page are not isolated examples. As we shall show, there exists a large number of closed formulae for the series \(\Omega _{m,n}\). By means of the linearization method (cf. [3, 4, 11, 12, 1618]), we shall reduce in the next section, for \(m,n\in \mathbb{Z}\), the series \(\Omega _{m,n}\) to \(\Omega _{m',0}\) with \(m'<0\). Then this last series will be evaluated in Sect. 3 via differential operators. The conclusive theorem affirms that, for all the \(m,n\in \mathbb{Z}\), the nonterminating \(\Omega _{m,n}\)-series can be always evaluated explicitly in terms of a finite number of algebraic functions in \(y_{\pm }\). Finally, by making use of Mathematica commands, 26 closed formulae are presented as exemplification.

2 Linearization method

By means of the linearization method, we shall establish, in this section, three reduction formulae that express ultimately the series \(\Omega _{m,n}\) with \(m,n\in \mathbb{Z}\) in terms of the series \(\Omega _{m',0}\), but with \(m'<0\).

2.1 \(m>0\)

By employing the Chu–Vandermonde formula on binomial convolutions, it is routine to prove the following linear representation lemma.

Lemma 1

(Linear representation)

For a natural number m and a variable y, the following linear relation holds:

$$\begin{aligned} \langle {y} \rangle _{m}=\sum_{i=0}^{m}(-1)^{i} \binom{m}{i} \langle {A+y} \rangle _{m-i}(A)_{i}. \end{aligned}$$

Now specifying in this lemma the parameters

$$\begin{aligned} y=k \quad\text{and}\quad A=3a-m+n-1, \end{aligned}$$

we get the equality

$$\begin{aligned} \langle {k} \rangle _{m}=\sum_{i=0}^{m}(-1)^{i} \binom{m}{i} \langle {3a-m+n-1+k} \rangle _{m-i}(3a-m+n-1)_{i}. \end{aligned}$$

By inserting this relation in the \(\Omega _{m,n}\)-series, we have

$$\begin{aligned} \Omega _{m,n}(a,x)={}&\sum_{k=0}^{\infty } \frac{(a)_{k}(a-\frac{1}{3})_{k}(a+\frac{1}{3})_{k}}{k!(\frac{1}{2}+m)_{k}(3a+n)_{k}}x^{2k} \\ ={}&\sum_{k=m}^{\infty } \frac{(a)_{k-m}(a-\frac{1}{3})_{k-m}(a+\frac{1}{3})_{k-m}}{(k-m)!(\frac{1}{2}+m)_{k-m}(3a+n)_{k-m}}x^{2k-2m} \\ &{}\times \sum_{i=0}^{m}(-1)^{i} \binom{m}{i} \frac{ \langle {3a-m+n-1+k} \rangle _{m-i}(3a-m+n-1)_{i}}{ \langle {k} \rangle _{m}}. \end{aligned}$$

Observing that

$$\begin{aligned} &\frac{ \langle {3a-m+n-1+k} \rangle _{m-i}}{(3a+n)_{k-m}}= \frac{(1-3a-n)_{2m}}{(3a-2m+n)_{k+i}}, \\ &\frac{(a)_{k-m}(a-\frac{1}{3})_{k-m}(a+\frac{1}{3})_{k-m}}{(k-m)! \langle {k} \rangle _{m}} =(-27)^{m} \frac{(a-m)_{k}(a-m-\frac{1}{3})_{k}(a-m+\frac{1}{3})_{k}}{k!(2-3a)_{3m}}; \end{aligned}$$

we can reformulate the double sum

Ω m , n ( a , x ) = ( 27 x 2 ) m ( 1 2 ) m ( 1 3 a n ) 2 m ( 2 3 a ) 3 m i = 0 m ( 1 ) i ( m i ) ( 3 a m + n 1 ) i ( 3 a 2 m + n ) i × k = m [ a m , a m 1 3 , a m + 1 3 1 , 1 2 , 3 a 2 m + n + i ] k x 2 k .

Expressing the last sum with respect to k in terms of \(\Omega _{0,m+n+i}(a-m,x)\), we derive the first reduction formula.

Proposition 2

(\(m\in \mathbb{N}_{0}\) and \(n\in \mathbb{Z}\))

$$\begin{aligned} \Omega _{m,n}(a,x) ={}& \biggl(-\frac{27}{x^{2}} \biggr)^{m} \frac{(\frac{1}{2})_{m}(1-3a-n)_{2m}}{(2-3a)_{3m}} \sum_{i=0}^{m}(-1)^{i} \binom{m}{i} \frac{(3a-m+n-1)_{i}}{(3a-2m+n)_{i}} \\ &{}\times \Biggl\{ \Omega _{0,m+n+i}(a-m,x) -\sum _{k=0}^{m-1} \frac{(3a-3m-1)_{3k}}{(2k)!(3a-2m+n+i)_{k}} \biggl( \frac{4x^{2}}{27} \biggr)^{k} \Biggr\} . \end{aligned}$$

2.2 \(n<0\)

Analogously, we can also prove, without difficulty, another linear representation lemma.

Lemma 3

(Linear representation)

For a negative integer n and a variable y, the following linear relation holds:

$$\begin{aligned} (A+y)_{-n}=\sum_{i=0}^{-n} \binom{-n}{i} \langle {B+y} \rangle _{i}(A-B+i)_{-n-i}. \end{aligned}$$

Under the parameter specification

$$\begin{aligned} y=k,\qquad A=3a+n,\qquad B=m-\frac{1}{2}, \end{aligned}$$

the equality in Lemma 3 can be restated as

$$\begin{aligned} (3a+n+k)_{-n}=\sum_{i=0}^{-n} \binom{-n}{i} \biggl\langle {m+k- \frac{1}{2}} \biggr\rangle _{i} \biggl(3a+n-m+\frac{1}{2}+i \biggr)_{-n-i}. \end{aligned}$$

By putting this relation inside the \(\Omega _{m,n}\)-series, we can manipulate the double sum

$$\begin{aligned} \Omega _{m,n}(a,x)={}&\sum_{k=0}^{\infty } \frac{(a)_{k}(a-\frac{1}{3})_{k}(a+\frac{1}{3})_{k}}{k!(\frac{1}{2}+m)_{k}(3a+n)_{k-n}}x^{2k} \\ &{}\times \sum_{i=0}^{-n} \binom{-n}{i} \biggl\langle {m+k-\frac{1}{2}} \biggr\rangle _{i} \biggl(3a+n-m+\frac{1}{2}+i \biggr)_{-n-i} \\ ={}&\sum_{i=0}^{-n} \binom{-n}{i} \biggl(3a+n-m+\frac{1}{2}+i \biggr)_{-n-i} \\ &{}\times \sum_{k=0}^{\infty } \biggl\langle {m+k- \frac{1}{2}} \biggr\rangle _{i} \frac{(a)_{k}(a-\frac{1}{3})_{k}(a+\frac{1}{3})_{k}}{k!(\frac{1}{2}+m)_{k}(3a+n)_{k-n}}x^{2k} \\ ={}&\sum_{i=0}^{-n}\binom{-n}{i} \frac{(\frac{1}{2}+m-i)_{i}(3a)_{n}}{(3a-m+\frac{1}{2})_{n+i}} \\ &{}\times \sum_{k=0}^{\infty } \frac{(a)_{k}(a-\frac{1}{3})_{k}(a+\frac{1}{3})_{k}}{k!(\frac{1}{2}+m-i)_{k}(3a)_{k}}x^{2k}. \end{aligned}$$

Writing the last sum by \(\Omega _{m-i,0}(a,x)\), we get the second reduction formula.

Proposition 4

(\(m,n\in \mathbb{Z}\) with \(n<0\))

$$\begin{aligned} \Omega _{m,n}(a,x)= \sum_{i=0}^{-n}(-1)^{i} \binom{-n}{i} \frac{(\frac{1}{2}-m)_{i}(3a)_{n}}{(3a-m+\frac{1}{2})_{n+i}} \Omega _{m-i,0}(a,x). \end{aligned}$$

2.3 \(n>0\)

The next linear relation comes from a limiting case of a known one. Dividing by \(A^{m}\) equation (3.1) in [17, Lemma 3.1] and then letting \(A\to \infty \), we get the following linearization lemma.

Lemma 5

(Linear representation)

For a natural number n and a variable y, the following linear relation holds:

$$\begin{aligned} 1=\sum_{i=0}^{n} \langle {B+y} \rangle _{n-i}(3C+3y)_{i} \mathrm{X}_{n}^{i}, \end{aligned}$$
(3)

where the coefficients \(\mathrm{X}_{n}^{i}\) are independent of the variable y and given explicitly by the two expressions

$$\begin{aligned} \mathrm{X}_{n}^{i} &=\sum_{j=0}^{i} \frac{(-1)^{n-i+j}}{i!} \binom{i}{j} \frac{3C-3B+3n-2i}{3(C-B+\frac{j}{3})_{n-i+1}} \\ &=\sum_{j=0}^{n-i} \frac{(-1)^{n-i+j}}{(n-i)!} \binom{n-i}{j} \frac{3C-3B+3n-2i}{(3C-3B+3j)_{i+1}}. \end{aligned}$$

Specifying in Lemma 5 the parameters

$$\begin{aligned} y=k,\qquad B=3a+n-1,\qquad C=a-\frac{1}{3}, \end{aligned}$$

the equality corresponding to (3) becomes

$$\begin{aligned} 1=\sum_{i=0}^{n} \langle {3a+n+k-1} \rangle _{n-i}(3a-1+3k)_{i} \mathcal{X}_{n}^{i} \end{aligned}$$
(4)

with the coefficients \(\mathcal{X}_{n}^{i}\) being determined by

$$\begin{aligned} \begin{aligned}[c] \mathcal{X}_{n}^{i} &=\sum_{j=0}^{i}\frac{(-1)^{n-i+j}}{i!} \binom{i}{j} \frac{2-6a-2i}{3(\frac{2}{3}-2a-n+\frac{j}{3})_{n-i+1}} \\ &=\sum_{j=0}^{n-i} \frac{(-1)^{n-i+j}}{(n-i)!} \binom{n-i}{j} \frac{2-6a-2i}{(2-6a-3n+3j)_{i+1}}. \end{aligned} \end{aligned}$$
(5)

By inserting this relation (5) in the \(\Omega _{m,n}\)-series, we get the double sum

Ω m , n ( a , x ) = k = 0 ( a ) k ( a 1 3 ) k ( a + 1 3 ) k k ! ( 1 2 + m ) k ( 3 a + n ) k x 2 k × i = 0 n 3 a + n + k 1 n i ( 3 a 1 + 3 k ) i X n i = i = 0 n ( 3 a 1 ) i ( 3 a + i ) n i X n i × k = 0 [ a + i 3 1 3 , a + i 3 , a + i 3 + 1 3 1 , 1 2 + m , 3 a + i ] k x 2 k .

Expressing the last sum by \(\Omega _{m,0}(a+\frac{i}{3},x)\), we have the third reduction formula.

Proposition 6

Let \(n\in \mathbb{N}\) and the connection coefficients \(\{\mathcal{X}_{n}^{i}\}\) be given by (5). Then the following formula holds:

$$\begin{aligned} \Omega _{m,n}(a,x) =\sum_{i=0}^{n} \mathcal{X}_{n}^{i} (3a-1)_{i}(3a+i)_{n-i} \Omega _{m,0} \biggl(a+\frac{i}{3},x \biggr). \end{aligned}$$

3 Conclusive theorem and examples

For a given integer pair \(\{m,n\}\), we can express the \(\Omega _{m,n}\)-series, by making use of Propositions 2, 4, and 6, in terms of \(\Omega _{m',0}\)-series with \(m'\le 0\). Therefore it remains to evaluate this last series. This will be done by utilizing differential operations. Suppose that \(f(x)\) is a differentiable function. Define the operator δ by

$$\begin{aligned} \delta f(x)=\frac{d}{dx} \biggl\{ \frac{f(x)}{x} \biggr\} . \end{aligned}$$

Then it is not hard to check that

δ Ω 0 , 0 ( a , x ) = k = 0 ( 2 k 1 ) [ a , a 1 3 , a + 1 3 1 , 1 2 , 3 a ] k x 2 k 2 = 1 x 2 Ω 1 , 0 ( a , x ) , δ 2 Ω 0 , 0 ( a , x ) = k = 0 ( 3 2 k ) [ a , a 1 3 , a + 1 3 1 , 1 2 , 3 a ] k x 2 k 4 = 3 x 4 Ω 2 , 0 ( a , x ) .

Proceeding by induction, we can show that

δ n Ω 0 , 0 ( a , x ) = ( 1 ) n 1 ( 2 n 3 ) ! ! k = 0 [ a , a 1 3 , a + 1 3 1 , 3 2 n , 3 a ] k ( 2 k 2 n + 1 ) x 2 k 2 n = ( 1 ) n ( 2 n 1 ) ! ! x 2 n Ω n , 0 ( a , x ) .

Recalling that

$$\begin{aligned} \Omega _{0,0}(a,x)=\frac{1}{2} \bigl\{ (1+y_{+})^{3a-1}+(1+y_{-})^{3a-1} \bigr\} \end{aligned}$$

and then relabeling n by −m, we get the following expression.

Proposition 7

For \(m<0\) and the three variables \(\{x,y_{\pm }\}\) related by (2), the following formula holds:

$$\begin{aligned} \Omega _{m,0}(a,x) =\frac{(-2/x^{2})^{m}}{2(\frac{1}{2})_{-m}} \delta ^{-m} \bigl\{ (1+y_{+})^{3a-1}+(1+y_{-})^{3a-1} \bigr\} . \end{aligned}$$

As an anonymous referee pointed out, instead of Proposition 4 the case \(n<0\) can be alternatively treated by repeatedly applying the operator δ to the initial function \(x^{6a-1}\Omega _{0,0}(a,x)\).

Summing up, for any given pair of integers m and n, the series \(\Omega _{m,n}(a,x)\) can be evaluated by carrying out the following procedure:

  • Step-A: If \(m>0\), write \(\Omega _{m,n}(a,x)\), by means of Proposition 2, in terms of \(\Omega _{0,n'}(a-m,x)\); then go to Step-B.

  • Step-B: For \(m\le 0\) and \(n\neq0\), apply Propositions 4 and 6 to express \(\Omega _{m,n}(a,x)\) as \(\Omega _{m',0}(a',x)\) with \(m'\le m\); then go to Step-C.

  • Step-C: Finally, for \(m\le 0\) and \(n=0\), evaluate \(\Omega _{m,0}(a,x)\), according to Proposition 7, by differentiating \(\Omega _{0,0}(a,x)\).

Therefore, we have shown the following general conclusion.

Theorem 8

For all the \(m,n\in \mathbb{Z}\), the nonterminating \(\Omega _{m,n}\)-series are always evaluable explicitly in a finite number of terms of algebraic functions in \(y_{\pm }\).

Based on Propositions 2, 4, 6, and 7, we have devised appropriately Mathematica commands that are employed to evaluate \(\Omega _{m,n}\) in closed forms for any specific integer pair “\(m,n\)”. Apart from the four formulae anticipated in the Introduction, we highlight further 26 elegant formulae as exemplification.

Example 1

(\(m=0\) and \(n=1\))

F 2 3 [ a , a + 1 3 , a 1 3 1 2 , 3 a + 1 | x 2 ] =w( y + )+w( y ),

where

$$\begin{aligned} w(y)=\frac{(1+y)^{3 a-1}}{2 (6 a+1)} \{1+6 a+y-3 a y \}. \end{aligned}$$

Example 2

(\(m=0\) and \(n=2\))

F 2 3 [ a , a + 1 3 , a 1 3 1 2 , 3 a + 2 | x 2 ] =w( y + )+w( y ),

where

$$\begin{aligned} w(y) &=\frac{(1+y)^{3 a-1}}{4 (2 a+1) (3 a+2) (6 a+1)} \begin{Bmatrix} 4+96 a^{2}+72 a^{3}+4 y+10 a y-42 a^{2} y \\ +38 a-72 a^{3} y-a y^{2}-3 a^{2} y^{2}+18 a^{3} y^{2} \end{Bmatrix}. \end{aligned}$$

Example 3

(\(m=0\) and \(n=-2\))

F 2 3 [ a , a + 1 3 , a 1 3 1 2 , 3 a 2 | x 2 ] =w( y + )+w( y ),

where

$$\begin{aligned} w(y) &=\frac{(1+y)^{3 a-1} (8-12 a-7 y+6 a y)}{(3 a-2)(y-2)^{3}}. \end{aligned}$$

Example 4

(\(m=0\) and \(n=-3\))

F 2 3 [ a , a + 1 3 , a 1 3 1 2 , 3 a 3 | x 2 ] =w( y + )+w( y ),

where

$$\begin{aligned} w(y) &=\frac{2(1+y)^{3 a-1}}{(a-1) (3 a-2) (2-y)^{5}} \begin{Bmatrix} 16-40 a+24 a^{2}-29 y+58 a y \\ -24 a^{2} y+15 y^{2}-19 ay^{2}+6 a^{2} y^{2} \end{Bmatrix}. \end{aligned}$$

Example 5

(\(m=1\) and \(n=0\))

F 2 3 [ a , a + 1 3 , a 1 3 3 2 , 3 a | x 2 ] =w( y + )+w( y ),

where

$$\begin{aligned} w(y) =\frac{(1+y)^{3 a-1}(4-12 a-5 y+6 a y)}{6 (1-2a)(6 a-5) y}. \end{aligned}$$

Example 6

(\(m=1\) and \(n=1\))

F 2 3 [ a , a + 1 3 , a 1 3 3 2 , 3 a + 1 | x 2 ] =w( y + )+w( y ),

where

$$\begin{aligned} w(y) &=\frac{-3(1+y)^{3 a-1}}{32y(3 a-\frac{5}{2})_{4} } \begin{Bmatrix} 8 a+24 a^{2}-144 a^{3}+5 y+4 a y-132 a^{2} y \\ +144 a^{3} y+5y^{2}-31 a y^{2}+60 a^{2} y^{2}-36 a^{3} y^{2} \end{Bmatrix}. \end{aligned}$$

Example 7

(\(m=1\) and \(n=-3\))

F 2 3 [ a , a + 1 3 , a 1 3 3 2 , 3 a 3 | x 2 ] =w( y + )+w( y ),

where

$$\begin{aligned} w(y) &= \frac{2 (1+y)^{3 a-1} (7-6 a-5 y+3 a y)}{3 y (a-1) (3 a-2) (y-2)^{3}}. \end{aligned}$$

Example 8

(\(m=-1\) and \(n=0\))

F 2 3 [ a , a + 1 3 , a 1 3 1 2 , 3 a | x 2 ] =w( y + )+w( y ),

where

$$\begin{aligned} w(y) =\frac{(1+y)^{3 a-1}}{2 (2-y)} \{2+y-6 a y \}. \end{aligned}$$

Example 9

(\(m=-1\) and \(n=1\))

F 2 3 [ a , a + 1 3 , a 1 3 1 2 , 3 a + 1 | x 2 ] =w( y + )+w( y ),

where

$$\begin{aligned} w(y) =\frac{1}{2} (1+y)^{3 a-1} \{1+y-3 a y \}. \end{aligned}$$

Example 10

(\(m=-1\) and \(n=2\))

F 2 3 [ a , a + 1 3 , a 1 3 1 2 , 3 a + 2 | x 2 ] =w( y + )+w( y ),

where

$$\begin{aligned} w(y) &=\frac{(1+y)^{3 a-1}}{4 (3 a+2)} \bigl\{ 4+6 a+4 y-6 a y-18 a^{2} y -3 a y^{2}+9 a^{2} y^{2} \bigr\} . \end{aligned}$$

Example 11

(\(m=-1\) and \(n=3\))

F 2 3 [ a , a + 1 3 , a 1 3 1 2 , 3 a + 3 | x 2 ] =w( y + )+w( y ),

where

$$\begin{aligned} w(y) &=\frac{(1+y)^{3 a-1}}{4 (a+1) (6 a+7)} \begin{Bmatrix} 14-16 a y-66 a^{2} y-36 a^{3} y-14 ay^{2}+30 a^{2} y^{2} \\ +26 a+12 a^{2}+14 y+36 a^{3} y^{2}+a y^{3}-9 a^{3} y^{3} \end{Bmatrix}. \end{aligned}$$

Example 12

(\(m=-1\) and \(n=-1\))

F 2 3 [ a , a + 1 3 , a 1 3 1 2 , 3 a 1 | x 2 ] =w( y + )+w( y ),

where

$$\begin{aligned} w(y) =\frac{(1+y)^{3 a-1}}{(2-y)^{3}} \bigl\{ 4-2 y-12 a y-3 y^{2}+6 a y^{2} \bigr\} . \end{aligned}$$

Example 13

(\(m=-1\) and \(n=-2\))

F 2 3 [ a , a + 1 3 , a 1 3 1 2 , 3 a 2 | x 2 ] =w( y + )+w( y ),

where

$$\begin{aligned} w(y) &=\frac{(1+y)^{3 a-1}}{(3 a-2) (y-2)^{5}} \begin{Bmatrix} 32-24 a y+144 a^{2} y+168 a y^{2}-144 a^{2}y^{2} \\ -48 a-48 y+35 y^{3}-72 a y^{3}+36 a^{2} y^{3} \end{Bmatrix}. \end{aligned}$$

Example 14

(\(m=2\) and \(n=-1\))

F 2 3 [ a , a + 1 3 , a 1 3 5 2 , 3 a 1 | x 2 ] =w( y + )+w( y ),

where

$$\begin{aligned} w(y) &=\frac{3(1+y)^{3 a-1}}{16y^{3}(3a-\frac{11}{2})_{4} } \begin{Bmatrix} 20-192 a y+72 a^{2} y+120 a y^{2} \\ -24 a+110 y-99 y^{2}-36 a^{2} y^{2} \end{Bmatrix}. \end{aligned}$$

Example 15

(\(m=2\) and \(n=-2\))

F 2 3 [ a , a + 1 3 , a 1 3 5 2 , 3 a 2 | x 2 ] =w( y + )+w( y ),

where

$$\begin{aligned} w(y) = \frac{3(1+y)^{3 a-1} (2+11 y-6 a y)}{4 y^{3}(2-3a) (3 a-\frac{11}{2})_{2}}. \end{aligned}$$

Example 16

(\(m=2\) and \(n=-3\))

F 2 3 [ a , a + 1 3 , a 1 3 5 2 , 3 a 3 | x 2 ] =w( y + )+w( y ),

where

$$\begin{aligned} w(y) &= \frac{6 (1+y)^{3 a-1} (1+5 y-3 a y)}{(3 a-3)_{2} (6 a-11) (y-2) y^{3}}. \end{aligned}$$

Example 17

(\(m=-2\) and \(n=0\))

F 2 3 [ a , a + 1 3 , a 1 3 3 2 , 3 a | x 2 ] =w( y + )+w( y ),

where

$$\begin{aligned} w(y) &=\frac{(1+y)^{3 a-1}}{6(2-y)^{3}} \begin{Bmatrix} 24-12 y-72 a y-14 y^{2}+y^{3} \\ +72 a y^{2}+72 a^{2} y^{2}-36a^{2} y^{3} \end{Bmatrix}. \end{aligned}$$

Example 18

(\(m=-2\) and \(n=1\))

F 2 3 [ a , a + 1 3 , a 1 3 3 2 , 3 a + 1 | x 2 ] =w( y + )+w( y ),

where

$$\begin{aligned} w(y) &=\frac{(1+y)^{3 a-1}}{2 (2-y)} \bigl\{ 2+y-6 a y-y^{2}+a y^{2}+6 a^{2} y^{2} \bigr\} . \end{aligned}$$

Example 19

(\(m=-2\) and \(n=-1\))

F 2 3 [ a , a + 1 3 , a 1 3 3 2 , 3 a 1 | x 2 ] =w( y + )+w( y ),

where

$$\begin{aligned} w(y) &=\frac{(1+y)^{3 a-1}}{(2-y)^{5}} \begin{Bmatrix} 16-24 y-48 a y+88 a y^{2}+48 a^{2} y^{2}+20 y^{3} \\ -16 ay^{3}-48 a^{2} y^{3}+y^{4}-8 a y^{4}+12 a^{2} y^{4} \end{Bmatrix}. \end{aligned}$$

Example 20

(\(m=-2\) and \(n=2\))

F 2 3 [ a , a + 1 3 , a 1 3 3 2 , 3 a + 2 | x 2 ] =w( y + )+w( y ),

where

$$\begin{aligned} w(y) =\frac{(1+y)^{3 a-1}}{4 (3a+2)} \begin{Bmatrix} 4+6 a+4 y-6 a y-18 a^{2} y \\ -5 a y^{2}+9 a^{2} y^{2}+18 a^{3}y^{2} \end{Bmatrix}. \end{aligned}$$

Example 21

(\(m=-2\) and \(n=3\))

F 2 3 [ a , a + 1 3 , a 1 3 3 2 , 3 a + 3 | x 2 ] =w( y + )+w( y ),

where

$$\begin{aligned} w(y) =\frac{(1+y)^{3 a-1}}{12 (a+1)} \begin{Bmatrix} 6+6 a+6 y-12 a y-18 a^{2} y-8 a y^{2} \\ +18 a^{2} y^{2}+18 a^{3} y^{2}+a y^{3}-9 a^{3} y^{3} \end{Bmatrix}. \end{aligned}$$

Example 22

(\(m=-3\) and \(n=1\))

F 2 3 [ a , a + 1 3 , a 1 3 5 2 , 3 a + 1 | x 2 ] =w( y + )+w( y ),

where

w(y)= ( 1 + y ) 3 a 1 10 ( 2 y ) 3 { 40 120 a y 30 y 2 + 132 a y 2 + 144 a 2 y 2 + 25 y 3 22 a y 3 20 y 180 a 2 y 3 72 a 3 y 3 5 y 4 a y 4 + 36 a 2 y 4 + 36 a 3 y 4 } .

Example 23

(\(m=-3\) and \(n=2\))

F 2 3 [ a , a + 1 3 , a 1 3 5 2 , 3 a + 2 | x 2 ] =w( y + )+w( y ),

where

w(y)= ( 1 + y ) 3 a 1 20 ( 3 a + 2 ) ( 2 y ) { 40 90 a y 180 a 2 y 20 y 2 24 a y 2 + 180 a 2 y 2 + 35 a y 3 + 60 a + 20 y + 216 a 3 y 2 33 a 2 y 3 180 a 3 y 3 108 a 4 y 3 } .

Example 24

(\(m=-3\) and \(n=3\))

F 2 3 [ a , a + 1 3 , a 1 3 5 2 , 3 a + 3 | x 2 ] =w( y + )+w( y ),

where

$$\begin{aligned} w(y) &=\frac{(1+y)^{3 a-1}}{20(a+1)} \begin{Bmatrix} 10-20 a y-30 a^{2} y-14 a y^{2}+30 a^{2}y^{2}+36 a^{3} y^{2} \\ +10 a+10 y+3 a y^{3}+2 a^{2} y^{3}-27 a^{3} y^{3}-18a^{4} y^{3} \end{Bmatrix}. \end{aligned}$$

Example 25

(\(m=3\) and \(n=-3\))

F 2 3 [ a , a + 1 3 , a 1 3 7 2 , 3 a 3 | x 2 ] =w( y + )+w( y ),

where

$$\begin{aligned} w(y) =\frac{-45 (1+y)^{3a-1}}{8y^{5}(3a-3)_{2}(3a-\frac{17}{2})_{4}} \begin{Bmatrix} 22+187 y-168 a y+36 a^{2} y+425 y^{2} \\ -12 a-575 ay^{2}+252 a^{2} y^{2}-36 a^{3} y^{2} \end{Bmatrix}. \end{aligned}$$

Example 26

(\(m=3\) and \(n=-2\))

F 2 3 [ a , a + 1 3 , a 1 3 7 2 , 3 a 2 | x 2 ] =w( y + )+w( y ),

where

w ( y ) = 45 ( 1 + y ) 3 a 1 32 y 5 ( 2 3 a ) ( 3 a 17 2 ) 5 × { 72 48 a 624 a y + 144 a 2 y + 1190 y 2 1916 a y 2 + 936 a 2 y 2 + 612 y 144 a 3 y 2 1105 y 3 + 1342 a y 3 540 a 2 y 3 + 72 a 3 y 3 } .

These identities are valid for all the x and y tied by (2) under the conditions \(|x|<1\) and \(-1/4< y<2\). When x is assigned to particular values, they may produce strange evaluation formulae. We limit ourselves to recording three groups of such formulae.

• Series with \(\{x, y_{+}, y_{-} \}= \{\sqrt{ \frac{3^{5}}{7^{3}}}, \frac{3}{4}, -\frac{2}{9} \}\).

figure c

• Series with \(\{x, y_{+}, y_{-} \}= \{2\sqrt{ \frac{3^{5}}{13^{3}}}, \frac{4}{9}, -\frac{3}{16} \}\).

figure d

• Series with \(\{x, y_{+}, y_{-} \}= \{5\sqrt{ \frac{3^{5}}{19^{3}}}, \frac{10}{9}, -\frac{6}{25} \}\).

figure e

To our knowledge, the formulae presented in this paper for \(\Omega _{m,n}(a,x)\) (when x is a free variable) have not appeared previously. Exceptions are about \(\Omega _{0,0}\), \(\Omega _{1,-1}\), and \(\Omega _{0,1}\). Their particular cases with \(\{x, y_{+}, y_{-}\}=\{1, 2, -1/4\}\) have been recorded by Milgram in his compendium [21, Equations 25, 30, 31]:

F 2 3 [ a , a + 1 3 , a 1 3 1 2 , 3 a | 1 ] = 3 3 a 1 ( 1 + 4 1 3 a ) 2 , F 2 3 [ a , a + 1 3 , a 1 3 3 2 , 3 a 1 | 1 ] = 3 3 a 1 ( 6 a 5 ) { 1 2 4 2 3 a } , F 2 3 [ a , a + 1 3 , a 1 3 1 2 , 3 a + 1 | 1 ] = 27 a 6 a + 1 { 1 2 + 9 a + 1 2 6 a + 1 } .

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References

  1. Bailey, W.N.: Some identities involving generalized hypergeometric series. Proc. Lond. Math. Soc. 29, 503–516 (1929)

    MATH  Google Scholar 

  2. Bailey, W.N.: Generalized Hypergeometric Series. Cambridge University Press, Cambridge (1935)

    MATH  Google Scholar 

  3. Chen, X., Chu, W.: Closed formulae for a class of terminating \(_{3}F_{2}(4)\)-series. Integral Transforms Spec. Funct. 28(11), 825–837 (2017)

    Article  MathSciNet  Google Scholar 

  4. Chen, X., Chu, W.: Terminating \(_{3}F_{2}(4)\)-series extended with three integer parameters. J. Differ. Equ. Appl. 24(8), 1346–1367 (2018)

    Article  Google Scholar 

  5. Choi, J.: Certain applications of generalized Kummer’s summation formulas for \(_{2}F_{1}\). Symmetry 13, Article ID 1538 (2021). https://doi.org/10.3390/sym13081538

    Article  Google Scholar 

  6. Chu, W.: Inversion techniques and combinatorial identities. Boll. Unione Mat. Ital., B 7(4), 737–760 (1993)

    MathSciNet  MATH  Google Scholar 

  7. Chu, W.: Inversion techniques and combinatorial identities: a quick introduction to hypergeometric evaluations. Math. Appl. 283, 31–57 (1994)

    MathSciNet  MATH  Google Scholar 

  8. Chu, W.: Binomial convolutions and hypergeometric identities. Rend. Circ. Mat. Palermo (Ser. II) 18, 333–360 (1994)

    MathSciNet  MATH  Google Scholar 

  9. Chu, W.: Generating functions and combinatorial identities. Glas. Mat. Ser. III 33, 1–12 (1998)

    MathSciNet  MATH  Google Scholar 

  10. Chu, W.: Some binomial convolution formulas. Fibonacci Q. 40(1), 19–32 (2002)

    MathSciNet  MATH  Google Scholar 

  11. Chu, W.: Analytical formulae for extended \(_{3}F_{2}\)-series of Watson–Whipple–Dixon with two extra integer parameters. Math. Compet. 81(277), 467–479 (2012)

    Article  Google Scholar 

  12. Chu, W.: Terminating \(_{4}F_{3}\)-series extended with two integer parameters. Integral Transforms Spec. Funct. 27(10), 794–805 (2016)

    Article  MathSciNet  Google Scholar 

  13. Gessel, I.M.: Finding identities with the WZ method. J. Symb. Comput. 20(5/6), 537–566 (1995)

    Article  MathSciNet  Google Scholar 

  14. Gessel, I.M., Stanton, D.: Strange evaluations of hypergeometric series. SIAM J. Math. Anal. 13, 295–308 (1982)

    Article  MathSciNet  Google Scholar 

  15. Gould, H.W.: Some generalizations of Vandermonde’s convolution. Am. Math. Mon. 63(1), 84–91 (1956)

    Article  MathSciNet  Google Scholar 

  16. Lewanowicz, S.: Generalized Watson’s summation formula for \(_{3}F_{2}(1)\). J. Comput. Appl. Math. 86, 375–386 (1997)

    Article  MathSciNet  Google Scholar 

  17. Li, N.N., Chu, W.: Nonterminating \(_{3}F_{2}\)-series with unit argument. Integral Transforms Spec. Funct. 29(6), 450–469 (2018)

    Article  MathSciNet  Google Scholar 

  18. Li, N.N., Chu, W.: Nonterminating \(_{3}F_{2}\)-series with a free variable. Integral Transforms Spec. Funct. 30(8), 628–642 (2019)

    Article  MathSciNet  Google Scholar 

  19. Maier, R.S.: A generalization of Euler’s hypergeometric transformation. Trans. Am. Math. Soc. 358, 39–57 (2005)

    Article  MathSciNet  Google Scholar 

  20. Maier, R.S.: The uniformization of certain algebraic hypergeometric functions. Adv. Math. 253, 86–138 (2014)

    Article  MathSciNet  Google Scholar 

  21. Milgram, M.: On hypergeometric \(_{3}F_{2}(1)\) – A review. Available at arXiv:1011.4546 [math.CA]; Updated version (2010)

  22. Riordan, J.: Combinatorial Identities. Wiley, New York (1968)

    MATH  Google Scholar 

  23. Zeilberger, D.: Forty “strange” computer–discovered and computer–proved (of course) hypergeometric series evaluations (2004). http://www.math.rutgers.edu/~zeilberg/ekhad/ekhad.html

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Li, N.N., Chu, W. Another class of nonterminating \(_{3}F_{2}\)-series with a free argument. Adv Differ Equ 2021, 496 (2021). https://doi.org/10.1186/s13662-021-03648-7

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