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Another class of nonterminating \(_{3}F_{2}\)-series with a free argument
Advances in Difference Equations volume 2021, Article number: 496 (2021)
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 [5–7, 13, 14, 19–21, 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:
where, according to Bailey [2, §2.1], the classical hypergeometric series reads as
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:
where the multiparameter notation for the former one will be abbreviated to
Our work is motivated by Lambert’s binomial series (see Riordan [22, §4.5] and [1, 8–10, 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
By the bisection of series, we have further four generating functions
Specifying with \(\beta =\frac{3}{2}\), making the replacements \(u\to \frac{2x}{3\sqrt{3}}, v\to y\), and then letting
respectively, in the above four equations, we get four hypergeometric formulae:

Here and forth, x and y are two variables related via equations
where \(y_{\pm }\) are computed from x through the fundamental algebraic relationship
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 “x–y curve” by rational functions:
The portions of the curve with \(t\in (-2,-1)\) and \(t\in (1,2)\) lie, in the “x–y 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.
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, 16–18]), 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:
Now specifying in this lemma the parameters
we get the equality
By inserting this relation in the \(\Omega _{m,n}\)-series, we have
Observing that
we can reformulate the double sum
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 \(n\in \mathbb{Z}\))
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:
Under the parameter specification
the equality in Lemma 3 can be restated as
By putting this relation inside the \(\Omega _{m,n}\)-series, we can manipulate the double sum
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 \(n<0\))
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:
where the coefficients \(\mathrm{X}_{n}^{i}\) are independent of the variable y and given explicitly by the two expressions
Specifying in Lemma 5 the parameters
the equality corresponding to (3) becomes
with the coefficients \(\mathcal{X}_{n}^{i}\) being determined by
By inserting this relation (5) in the \(\Omega _{m,n}\)-series, we get the double sum
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:
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
Then it is not hard to check that
Proceeding by induction, we can show that
Recalling that
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:
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 \(n=1\))
where
Example 2
(\(m=0\) and \(n=2\))
where
Example 3
(\(m=0\) and \(n=-2\))
where
Example 4
(\(m=0\) and \(n=-3\))
where
Example 5
(\(m=1\) and \(n=0\))
where
Example 6
(\(m=1\) and \(n=1\))
where
Example 7
(\(m=1\) and \(n=-3\))
where
Example 8
(\(m=-1\) and \(n=0\))
where
Example 9
(\(m=-1\) and \(n=1\))
where
Example 10
(\(m=-1\) and \(n=2\))
where
Example 11
(\(m=-1\) and \(n=3\))
where
Example 12
(\(m=-1\) and \(n=-1\))
where
Example 13
(\(m=-1\) and \(n=-2\))
where
Example 14
(\(m=2\) and \(n=-1\))
where
Example 15
(\(m=2\) and \(n=-2\))
where
Example 16
(\(m=2\) and \(n=-3\))
where
Example 17
(\(m=-2\) and \(n=0\))
where
Example 18
(\(m=-2\) and \(n=1\))
where
Example 19
(\(m=-2\) and \(n=-1\))
where
Example 20
(\(m=-2\) and \(n=2\))
where
Example 21
(\(m=-2\) and \(n=3\))
where
Example 22
(\(m=-3\) and \(n=1\))
where
Example 23
(\(m=-3\) and \(n=2\))
where
Example 24
(\(m=-3\) and \(n=3\))
where
Example 25
(\(m=3\) and \(n=-3\))
where
Example 26
(\(m=3\) and \(n=-2\))
where
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 \(\{x, y_{+}, y_{-} \}= \{\sqrt{ \frac{3^{5}}{7^{3}}}, \frac{3}{4}, -\frac{2}{9} \}\).

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

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

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]:
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The authors express their sincere gratitude to the two anonymous referees for generous comments and valuable suggestions that have significantly contributed to improving the manuscript during the revision.
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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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DOI: https://doi.org/10.1186/s13662-021-03648-7
MSC
- 33C20
- 05A10
Keywords
- Classical hypergeometric series
- Linearization method
- Bisection series
- Nonterminating \(_{3}F_{2}\)-series
- Lambert’s binomial series