1. Introduction and Outline
For an indeterminate
x and
, the Pochhammer symbol is defined by
It can be expressed in terms of the
-function
For the sake of brevity, the
-function quotient will be abbreviated to
Denote the Euler constant by
. Then, the logarithmic differentiation of the
-function results in the digamma function (cf. Rainville [
1], §9)
Let
stand for the coefficient of
in the formal power series
. For a real number
, we can extract the coefficients
from the exponential expression
where the Riemann and Hurwitz zeta functions are defined, respectively, by
For
and
, define the parametric harmonic numbers by
When
and
, they reduce to the usual harmonic numbers
In case
, it will be suppressed from these notations. We record also the following simple, but useful, relations:
The parametric harmonic number of the first order can be obtained by extracting the coefficient from the factorial quotient
By means of the generating function method, it can be shown without difficulty that in general, there hold the following formulae:
Here, the Bell polynomials (cf. [
2], §3.3) are expressed by the multiple sum
and
is the set of
m-partitions represented by
m-tuples
subject to the condition
. We sketch proofs of (
2) for integrity. According to power series expansion of the logarithm function
we can manipulate the two factorial quotients
Then the formulae in (
2) follow by extracting the coefficient of
across the above two equations.
There exist numerous infinite series representations for
and related mathematical constants in the literature (cf. [
3,
4,
5,
6,
7,
8,
9,
10,
11]). Some of them can be shown by means of the hypergeometric series approach. Even though many summation formulae (cf. [
12,
13,
14]) have been found for the hypergeometric series, these with a half argument are quite rare. The two fundamental ones are due to Gauss (called Gauss’ second summation theorem) and Bailey (cf. [
12], §2.4):
Following a recent work of the second author [
15], we shall investigate infinite series involving harmonic numbers, by examining primarily Gauss’ Formula (
4) and secondarily Bailey’s Formula (
5). Several remarkable identities will be established, including eight conjectured ones made experimentally by Sun [
16,
17]. This will be fulfilled by utilizing the “coefficient extraction method” (cf. [
5,
18,
19]). Considering that for a harmonic series of convergence rate “
”, there are only a few existing formulae scattered in the literature up to now, so the relatively full coverage presented in this paper may serve as a reference source for readers.
In the next section, 17 infinite series with central binomial coefficients in numerators will be evaluated in closed form by exploring four cases of Gauss’ second summation Formula (
4). Then, a further 23 infinite series identities with central binomial coefficients in denominators will be derived, in
Section 3, by examining two cases of (
4) and one case of Bailey’s Formula (
5).
In order to ensure the accuracy of our computations, numerical tests for all the equations have been made by appropriately devised Mathematica commands.
2. Series with Binomial/Multinomial Coefficients in Numerators
By examining Gauss’ second summation formula (
4), we shall evaluate, in closed form, several infinite series involving harmonic numbers, and binomial/multinomial coefficients in numerators, including five conjectured series made recently by Sun [
17].
2.1.
The corresponding series becomes
Both members of the above equation are analytic function of
x in the neighborhood of
and can be expanded into power series in
x. Denoting by
the coefficient of
across the equation, we derive the following infinite series identities.
First, letting
, the resulting coefficient
evaluates an easier series:
The next coefficient
gives rise to the identities below.
Proof. The first identity (a) can be found in [
15] (§3.1), By comparing the coefficient
, we deduce the identity
Then, by putting the above equality in conjunction with (a), we confirm the second identity (b), which is equivalent to the following one:
conjectured recently by Sun [
17] (Equation (2.18)). □
By examining the two coefficients
and
, we can evaluate the following two series on quadratic harmonic numbers, where the first one was conjectured by Sun [
17] (Equation (2.19)).
In general, the series corresponding to coefficients of higher powers of x become more complicated. For instance, from and , we have two further formulae.
2.2.
The corresponding series becomes
Both members of the above equation are analytic function of
x in the neighborhood of
and can be expanded into power series in
x. Denoting by
the coefficient of
across the equation, we derive the following infinite series identities.
The initial coefficient
yields the following formula:
Proof. The identity (a) can be found in [
15] (§4.1). By considering the coefficient
, we have the next formula
Then, combining this one with (a) leads us to identity (b). The two formulae in this theorem can be considered as refinements of the following conjectured identity made by Sun [
17] (Equation (2.20)):
□
Two further series can be evaluated by examining the coefficients and as in the following proposition.
2.3.
The corresponding series becomes
Both members of the above equation are analytic functions of
x in the neighborhood of
and can be expanded into power series in
x. Denoting by
the coefficient of
across the equation, we derive the following infinite series identities.
The initial coefficient
results in the following identity:
Then, we have three independent series related to
.
Proof. For the first series (a), refer to Chu [
15] (§4.1). By examining the coefficient
, we obtain the next formula
By combining this with (a), we derive (b), which was conjectured by Sun [
17] (Equation (2.22)). Finally the identity (c) corresponds to the coefficient
. □
Analogously, two further identities can be shown by considering and .
2.4.
The corresponding series becomes
Both members of the above equation are analytic function of
x in the neighborhood of
and can be expanded into power series in
x. Denoting by
the coefficient of
across the equation, we derive the following infinite series identities.
The initial coefficient
gives rise to the formula below.
Then, we have two independent series evaluations.
Proof. The first series (a) is due to the second author [
15] (§4.1). According to the coefficient
, we can derive another formula
Then by combining (a) with the above identity, we find the identity (b), which was conjectured by Sun [
17] (Equation (2.27)). □
3. Series with Central Binomial Coefficients in Denominators
By employing Gauss’ second theorem (
4) and then Bailey’s theroem (
5), we shall establish several summation formulae concerning harmonic numbers, and central binomial coefficients in denominators. Three of them were previously conjectured (without proofs) by Sun [
16] (2014).
3.1.
The corresponding series in (
4) becomes
Both members of the above equation are analytic function of
x in the neighborhood of
and can be expanded into power series in
x. Denoting by
the coefficient of
across the equation, we derive the following infinite series identities.
The summation formula corresponding to the coefficient
reads as
By examining the coefficients
and
, we derive the next two identities, where the former one was conjectured by Sun [
16] (Equation (1.14)).
We have also two further summation formulae of infinite series.
Proof. The first identity (a) follows directly by extracting coefficient . Instead, the identity (b) is confirmed by evaluating the limit of as and . □
3.2.
The corresponding series in (
4) becomes
Letting
be the coefficient of
across the equation, we can derive the following infinite series identities. The initial one related to
is given by
Next, by considering
,
and
, we can derive the following three elegant summation formulae, where (b) can be found in Sun [
20].
Two further identities are recorded in the next proposition, that are deduced by the limiting case of , and the coefficient , respectively.
3.3. Bailey’s -Series
Under the replacements
and
, we can reformulate Bailey’s formula (
5) as
Both members of the above equation are analytic function of
x in the neighborhood of
and can be expanded into power series in
x. Denoting by
the coefficient of
across the equation, we derive the following infinite series identities, which may serve as complementary results to those obtained recently by the second author [
15]. Since there are more summation formulae in this subsection, we exhibit them in groups according to the similarity of their summands.
First, by considering the coefficients and , we immediately obtain two identities, where G is the Catalan constant as usual.
Then we have closed formulae below for three independent series.
We remark that this theorem refines the following two identities conjectured by Sun [
16] (Equations (1.15) and (1.16)):
Proof. The first identity (a) is easiest, which follows simply by extracting the coefficient . Instead, the second identity (b) is not deducible by hypergeometric series approach. We offer the following integration proof.
Recalling the power series expansion
we can manipulate the integrals
By making use of the Fourier series expansion
we can evaluate the integral
By making substitution and simplifications, we find the identity (b).
Finally, the identity (c) follows from a linear combination of (a), (b) as well as that displayed in Theorem 6 (a). □
Now, we give three pairs of similar summation formulae. The first pair of series are evaluated in closed form by extracting the coefficient and . □
The following two similar series are evaluated in closed form by extracting the coefficients and .
By examining the coefficients and , we establish two identities as in the next theorem.
Finally we record three seemingly unrelated series, which correspond to the coefficients , and , respectively.