A New Closed-Form Formula of the Gauss Hypergeometric Function at Specific Arguments

: In this paper, the authors briefly review some closed-form formulas of the Gauss hyperge-ometric function at specific arguments, alternatively prove four of these formulas, newly extend a closed-form formula of the Gauss hypergeometric function at some specific arguments, successfully apply a special case of the newly extended closed-form formula to derive an alternative form for the Maclaurin power series expansion of the Wilf function, and discover two novel increasing rational approximations to a quarter of the circular constant

The logarithmic derivative [ln Γ(z)] ′ = Γ ′ (z) Γ(z) is denoted by ψ(z) and is called the psi function or the digamma function.The reciprocal 1  Γ(z) is an entire function possessing simple zeros at the points 1 − k for k ∈ N (see [5] We note that the definition (2) of B(z, w) extends the following classical definition: z+w dt, ℜ(z), ℜ(w) > 0.

A Brief Review
In general, it is not easy to write out elementary, closed-form, explicit expressions of the Gauss hypergeometric function 2 F 1 (a, b; c; z) at specific arguments (a, b; c; z).See the short and simple review in [6] (Section 4).
In the paper [7], the authors reviewed many results obtained in the papers [8][9][10] and other historical literature about the generalized hypergeometric series p F q .In the recently published papers [11][12][13][14], the authors derived more significant conclusions for p F q at some specific arguments.
In the paper [7], among other things, Rakha and Rathie established several closed-form formulas of the Gauss hypergeometric function 2 F 1 (a, b; c; z) for z = −1, 1 2 as follows.1.
In [22] (Lemma 2.6), for 0 ̸ = |t| < 1 and n ∈ N, Qi successfully discovered and applied the closed-form formula where i = √ −1 is the imaginary unit.In [22] (Remark 6.6), Qi conjectured that the range of n ∈ N in (15) can be extended to n ∈ R.This conjecture still remains open at present.See also [23] (Section 3.9).
In Section 3 of this paper, we will alternatively compute four Gauss hypergeometric function: In Section 4 of this paper, more importantly, we will extend the closed-form Formula ( 16) by establishing a closed-form expression of the Gauss hypergeometric function for n ∈ N 0 and z ∈ C. In Section 5, we will apply a special case of the newly extended closed-form formula for the function (18) to derive an alternative form for the Maclaurin power series expansion of the Wilf function which was investigated in the conference paper [25] and the preprints on the site https: //arxiv.org/abs/2110.08576(accessed on 1 May 2022); moreover, we will discover two novel increasing rational approximations to the irrational constant π 4 .In the final section, Section 6, we will list some more remarks on our main results and related findings.

Alternative Proofs of Four Known Results
Now, we set out to alternatively compute the four Gauss hypergeometric function listed in (17).
Proof.The alternative proof of ( 5) is as follows.Taking z = −1 and c = a − b + 2 in (24) and employing ( 3) and ( 4) lead to The explicit Formula ( 5) is thus alternatively proved.
Applying Equation ( 30), we can derive more explicit formulas of the Gauss hypergeometric function 2 F 1 (a, b; c; z) at specific arguments, as follows.

A New Closed-Form Formula
In this section, we start off to derive a closed-form formula for the specific Gauss hypergeometric function in (18).This result generalizes the closed-form Formula ( 16), which was established by Qi in [24] (Corollary 4.1).Theorem 3.For n ∈ N 0 , we have where Proof.In [4] (p. 109, Example 5.1), it is given that By virtue of Abel's limit theorem in [26] (p.245, Theorem 9.31), we can take z = 1 in (41) and obtain This can also be derived from ( 14) by taking a = 1 2 , b = 1, and j = 1.In [5] (p.556, Entry 15.1.8),the formula and, by virtue of Abel's limit theorem in [26] (p.245, Theorem 9.31), This can also be derived from ( 14) by taking a = 3 2 , b = 1, and j = 0. Theorem 1.1 in the paper [27] reads that, for any integers k, ℓ, and m, there are unique functions P k,ℓ,m (a, b; c; z) and Q k,ℓ,m (a, b; c; z), rational in the parameters a, b, c, and z, with In particular, letting k = ℓ = m = n ∈ N 0 , setting (a, b, c) = 1 2 , 1, 3 2 , replacing z by −z 2 in (44), making use of Formula (41), and replacing z by −z 2 in (43) all yield the following: where ; −z 2 are rational in the parameter z 2 , with From [16] (p.388, Entry 15.5.19),we obtained Replacing z by −z 2 and letting (a, b, c) In [5] (p.556, Entry 15.1.10),the formula 2 F 1 a, is obtained.Setting a = 3 2 and replacing z by z i in (49) result in (50) In [5] (p.558, Entry 15.2.20), the formula 3  2 , replacing z by −z 2 in (51), and employing (50) reveal (52) In [5] (p.558, Entry 15.2.11), the formula 3  2 , we replace z by −z 2 in (53) and utilize Formula (41), which lead to Substituting (54) into (52) and then simplifying the result yield the following: (55) This means that Taking n = 0 in (48), utilizing (41) and (55), and then reformulating these formulas allow us to determine This means that Through similar arguments as those above, taking n = 1, 2 in (48) and considering the explicit Formulas (56) and (57), we repeatedly derive Then, we guess that rational functions P n z 2 and Q n z 2 defined in (45) should be (39) and where we assume c 0,0 = c 1,0 = 1 and an empty sum is understood to be zero.We also guess that the numbers c n,k for 0 ≤ k ≤ n − 1 and n ∈ N are positive integers.We list the first few values of the coefficients c n,k for 0 ≤ k ≤ n − 1 and 1 ≤ n ≤ 8 in Table 1, which were announced by Qi on the website https://mathoverflow.net/q/436464/ (accessed on 27 March 2024) as a problem.Substituting (45) into (48) and then simplifying the outcome yield for n ∈ N.This can be written as two recurrent relations and for n ∈ N, with initial values in (46), ( 56)-(58).Substituting (39) into (60) and then simplifying result in for n ∈ N.This equality can be straightforwardly verified to be true.As a result, Formula (39) is valid for n ∈ N 0 .Substituting (59) into (61) and then simplifying result in Using the fact that which is a direct consequence of the definition (62), and equating the coefficients of z 2k for 0 and for 2 ≤ k ≤ n and n ∈ N.
The identity (75) follows from comparing the closed-form Formulas ( 16) and ( 74) and from simplification.Remark 3. Since the expression (72) is an alternating sum, we cannot directly confirm the positivity of the rational sequence C n,k from its appearance.
From the explicit Formula (72), we cannot clearly see what the closed-form formula of the sequence c n,k for 0 ≤ k ≤ n − 1 and n ∈ N is, nor can we clearly see whether the numbers c n,k for 0 ≤ k ≤ n − 1 and n ∈ N are positive integers.

The Third Problem by Wilf and Rational Approximations
The third problem posed by Herbert S. Wilf (1931-2012) on the site https://www2 .math.upenn.edu/~wilf/website/UnsolvedProblems.pdf(accessed on 26 July 2021) states that, if the function W(z) defined in (19) has the Maclaurin power series expansion find the first term of the asymptotic behaviour of the a n 's.
In the conference paper [25], Ward considered this problem.We now recite the texts of the review https://mathscinet.ams.org/mathscinet-getitem?mr=2735366 (accessed on 1 August 2021) by Tian-Xiao He for the paper [25] as follows.
The coefficient a n can be written as where b n and c n are non-negative rational numbers.In fact, and the rational numbers of the form c n b n provide approximations to π.A complete expansion of the coefficients a n is found by the author.It is probably the best that can be performed, given the oscillatory nature of the terms.
Wilf's comments on the paper [25] on 13 December 2010 is quoted as follows: "Mark Ward has found a complete expansion of these coefficients.It's not quite an asymptotic series in the usual sense, but it is probably the best that can be done, given the oscillatory nature of the terms." In the preprints on the site https://arxiv.org/abs/2110.08576(accessed on 1 May 2022), among other findings, Qi discovered Formula ( 16) and expanded the Wilf function for |z| < ln 2, where the Stirling numbers of the second kind S(n, k) for n ≥ k ≥ 0 can be analytically generated (see [29] (p.51) and [30]) by According to the notations used in (76), the Maclaurin power series expansion (77) can be alternatively expressed as for n ∈ N.
The sequence 4n!b n for n ≥ 0 is positive, increasing, and logarithmically convex; 2.
The limits Making use of the equality (75), we can reformulate the sequence c n in (78) as Employing (80), we can rewrite the Maclaurin power series expansion (77) as for |z| < ln 2. As a result, we derive an alternative form (81) for the Maclaurin power series expansion of the Wilf function W(z) defined by (19).The third limit in (79) can be explicitly formulated as Motivated by the identity (74) and the difference in the parentheses in (81), stimulated by numerical computation, and hinted by the limit (82) and the Stolz-Cesàro theorem for calculating limits, we guess that the sequences are increasing in k ∈ N and tend to π 4 as k → ∞.This guess was also posted on the site https://math.stackexchange.com/q/4883527(accessed on 19 March 2024).
Perhaps it is difficult to directly verify the above guess.However, we find out a simple proof of the above guess as follows.
Theorem 4. The rational sequences in (83) are increasing in k ∈ N and tend to the irrational constant π 4 as k → ∞.
Proof.The Euler integral representation of the Gauss hypergeometric function 2 F 1 (see [15] (p.66, Theorem 2.2.1) and [31] in the x plane cut along the real axis from 1 to ∞, where it is understood that arg t = arg(1 − t) = 0 and (1 − xt) −a has its principle value.Setting (a, b; c; x) = n + 1 2 , n + 1; n + 3 2 ; −1 , n ∈ N 0 in (84) and simplifying give for n ∈ N 0 .Hence, we obtain for n ∈ N 0 , which is decreasing in n ∈ N 0 and tends to 0 as n → ∞.Combining the integral representation (85) with Formula (74) reveals which is increasing in n ∈ N 0 and tends to π 4 .The proof of Theorem 4 is thus complete.

More Remarks
In this section, we list more remarks on our main results and related ones.
From (88), it follows that x n e (1−v)x (e x −z) 2 dx > 0 for n ∈ N 0 , v > 0, and z ∈ (−∞, 1).This means that, for any fixed real number z ∈ (−∞, 1), the real function is completely monotonic with respect to the variable v ∈ (0, ∞).For details about completely monotonic functions, please refer to the review article [35] and closely related references therein.
Remark 5. On the site https://mathoverflow.net/q/423800 (accessed on 30 March 2023), Qi asked the question: can one find an elementary function f (t) such that Taking a = 1 2 , b = 1 2 , and c = 1 in (90) yields In [4] (p. 128), we can find two relations for |t| < 1 between the Gauss hypergeometric function 2 F 1 and the complete elliptic integrals of the first and second kinds K(t) and E(t).Substituting two formulas in (92) into (91) gives Formula (93) reveals that the Gauss hypergeometric function 2 F 1 1 2 , 1 2 ; 2; t for |t| < 1 should not be an elementary function.
The above question with its motivation and the above answer were mentioned in [6] (Section 4.2).Remark 6.As a continuation of the question (89) and the answer by Gerald A. Edgar on the site https://mathoverflow.net/a/423802 (accessed on 2 June 2022), Qi asked an alternative question on https://math.stackexchange.com/q/4669567(accessed on 30 March 2023) which can be revised and quoted as follows.
Can one write out a closed-form formula for the general term of the coefficients in the Maclaurin power series expansion of the power function In other words, is there a closed-form expression for the coefficients C m,n in the power series expansion The intention of this question is the same one as that stated in [6] (Section 4.3).
As performed in the proof of [6] (Theorem 1), we can derive a recursive relation for the coefficients C m,n .However, we are more interested in a possible closed-form formula for the coefficients C m,n .Remark 7. On the site https://mathoverflow.net/q/448555 (accessed on 15 June 2023), Qi asked the following two questions: 1.
On the site https://mathoverflow.net/a/458325 (accessed on 13 November 2023), Gerald A. Edgar (Ohio State University, USA) wrote that, when taking a = was expanded into a Maclaurin power series at x = 0. • In [38] (Theorem 2), among other findings, the function ln CosR n (x) for n ≥ 2 in (100) was proven to be decreasing and concave on 0, π 2 .These results are weaker than the corresponding ones in [36] (Theorem 2), not only because a positive concave function must be a logarithmically concave function (but the converse is not true), but also because we consider the including relations 0, π 2 ⊂ (0, ∞) and 0, π 2 ⊂ (0, π).