Ramanujan-inspired series for 1/π involving harmonic numbers

ABSTRACT By applying the derivative operator to the known identities from hypergeometric series or WZ pairs, we obtain six series associated with harmonic numbers. Specifically, five of them are Ramanujan-like formulas for $ 1/\pi $ 1/π and the remaining one contains harmonic numbers of order 2. As conclusions, Sun's four conjectural series are proved.


Introduction
In 1914, Ramanujan [15] first systematically investigated some series for 1/π , including the following beautiful ones Here and in what follows, the Pochhammer symbol is defined as where (x) is Euler's gamma function and n is a nonnegative integer.
In 2013, Guillera [14, (32)] proved an elegant Ramanujan-type formula where H n denotes the ordinary harmonic numbers For a complex number x and a positive integer l, the generalized harmonic numbers of order l are defined as In [16], Sun conjectured hundreds of series and congruences with summands containing harmonic numbers, such as which was recently confirmed by Wei [17] through the derivative operator method.It is an interesting topic to evaluate summations involving harmonic numbers.Apart from [14] and [16], more mathematical literature concerning harmonic sums can be found in [1, 4-8, 13, 18, 19].Motivated by the work forementioned and the formulas (1)-(3), we establish six Ramanujan-type representations for 1/π involving harmonic numbers as follows.We see that Sun's three conjectural series are verified.

Theorem 1.6 ([16, Equation (4.37)]):
As a prerequisite, we review some basic concepts.Given a differentiable function f (x), the derivative operator D x is defined as Besides, the digamma function is given by where γ stands for the Euler's constant.By acting the operator D x on the above equation, we obtain Additionally, there is a recurrence relation about the digamma function and some concrete values which will be utilized in the proof The rest of this paper is organized as follows.By means of the derivative operator method, Theorems 1.1-1.6 shall be proved in Sections 2 -5 respectively.We will also employ WZ ( Wilf-Zeilberger) pairs to derive Theorem 1.5.

Proof of Theorem 1.1
In fact, Theorem 1.1 has been obtained by Campbell [5,Theorem 5].Here, we provide another way to its proof.In the proof of Theorem 1.1, Dougall's well-poised 5 F 4 -series (cf. [3, P. 27]) counts for much where Notice that ( 12) has also been showed by Ekhad and Zeilberger [10] using the WZ method.
Draw the operator D c on both sides of (12) to attain Letting c = 1 2 in the above equation, we arrive at Theorem 1.1.

Proofs of Theorems 1.2-1.4
The hypergeometric transformation formula due to Chu and Zhang [9, Theorem 9] may contribute to the derivations of Theorems 1.2-1.4: where 13) and use Dougall's theorem (11) to acquire For a positive integer n, taking e = −n → −∞ in (14) and observing the property [2, Equation (1.4.3)] we are led to Apply the operator D c on both sides of (15) to gain The c = 1 2 case of ( 16) produces the desired result (6).
Proof of Theorem 1.3: where Adopt the operator D b on both sides of (17) to draw The b = 1 2 case of (18) gives The combination of ( 6) and ( 19) yields the wanted result (7).
Proof of Theorem 1.4: Substitute (a, c, d, e) = 1 2 , 1 2 , 1 4 , 3  4 in (13) and multiple both sides by 1  2 − b to get where Utilize the operator D b on both sides of (20) to obtain Based on the formula Opt (a, b, d, e) = 1 2 , 1 2 , 1 4 , 3  4 in (13) to get where Employ the operator D c on both sides of (23) to gain The c → 1 2 − case of (24) becomes where Apply the operator D d on both sides of (26) to acquire The Replace (a, b, c, d) = (1 − e) 5  4 − e where Conduct the operator D e on both sides of (29) to arrive at The e → 3 4 − case of (30) concludes The result of the linear combination (25)-( 22)-( 28)-(31) coincides with (8) completely.Hence, we finish the proof of this theorem.

Proof of Theorem 1.5
Different from the previous proofs, in this section, we will carry out the derivative on the identity that is formulated by WZ pairs.By setting up suitable WZ pairs (F(n, k), G(n, k)), Guillera [12] showed that several identities involving π can be extended to the ones with an extra parameter.For example, for such WZ pairs Guillera [12] obtained, for any k, In (32), taking the derivative with respect to k and then setting k = 0, we derive that Therefore, by (3) and the above equation, we prove the truth of Theorem 1.5.

Proof of Theorem 1.6
Recall that another hypergeometric transformation [9, Theorem 14] is stated as where where Carry out the operator D b on both sides of (34) to get Divide both sides of (35) by 1−2b to get The b → 1 (40) Hence, the linear combination 8 × (37) + 3 × (40) is just the result (10).