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A note on some reduction formulas for the incomplete beta function and the Lerch transcedent

We derive new reduction formulas for the incomplete beta function and the Lerch transcendent in terms of elementary functions. As an application, we calculate some new integrals. Also, we use these reduction formulas to test the performance of the algorithms devoted to the numerical evaluation of the incomplete beta function.

Nevertheless, reduction formulas for B (ν, 0, z) when ν is a rational number do not seem to be reported in the most common literature. The aim of this note is just to provide such reduction formulas in terms of elementary functions. As an application, we will calculate some new integrals in terms of elementary functions. Also, we will check that the numerical evaluation of the incomplete beta function is improved with these reduction formulas.
This paper is organized as follows. Section 2 derives reduction formulas for B (ν, 0, z), both for ν positive rational, as well as negative rational. Particular cases for ν non-negative integer or ν half-integer are also derived. In Section 3, we will apply the reduction formulas derived in Section 2 to calculate some integrals which do not seem to be reported in the most common literature. Further, for particular values of the parameters, the symbolic computation of these integrals is quite accelerated by using the aforementioned reduction formulas. Also, we will use these reduction formulas to numerically test the performance of the algorithm provided in MATHEMATICA to compute the incomplete beta function.
Theorem 1 For ν = n + p q ∈ Q + , with n = ⌊ν⌋ and p < q, the reduction formula holds true.
Remark 1 Notice that the reduction formula (9) is included in (13), but not (7), which is a singular case.

Applications
From the reduction formulas obtained in Section 2, next we calculate some integrals in terms of elementary functions. Also, we will use these reduction formulas as a benchmark for the computation of the incomplete beta function.
Therefore, from (13) and (19), we have for ν = n + p q ∈ Q + , with n = ⌊ν⌋ and p < q, On the other hand, in the literature we found [7, Eqn. 58:14 Therefore, from (13) and (21), we have for λ = n + p q ∈ Q + , with n = ⌊λ⌋ and p < q, The integral given in (22) generalizes the results found in the literature for λ = n + 1 and λ = n + 1 2 with n = 0, 1, 2, . . . [3,. It is worth noting that for particular values of λ, the Integrate MATHE-MATICA command is able to compute symbolically the same results as (22), but in a very time-consuming way. For instance, for λ = 5 4 , we obtain but the Integrate command takes around 300 times longer than the reduction formula given in (22).

Numerical evaluation
From a numerical point of view, the reduction formulas (13) and (16) are quite useful to plot B (ν, 0, z) as a function of ν in the real domain. However, for some real values of ν and z, we obtain a complex value for B (ν, 0, z). In these cases, the imaginary part of B (ν, 0, z) is not always easy to compute. Figure 1 shows the plot of Im (B (ν, 0, z)) as a function of z for ν = 12.3. The reduction formula (13) shows the correct answer, i.e. Im (B (ν, 0, z)) = −π, meanwhile the numerical evaluation of Im (B (ν, 0, z)) with MATHEMATICA diverges from this result. A similar feature is observed using (16) and a negative value for ν. It is worth noting that the equivalent numerical evaluation of Im (z ν Φ (z, 1, ν)) with MATHEMATICA yields also −π.

Conclusions
On the one hand, we have derived in (13) and (16) new expressions for the incomplete beta function B (ν, 0, z) and the Lerch transcendent Φ (z, 1, ν) in terms of elementary functions when ν is rational and z is complex. Particular formulas for non-negative integers values of ν and for half-integer values of ν are given in (7) and (9), (15) respectively. On the other hand, we have calculated the integrals given (20) from the reduction formulas (13) and (16) and the integral representation of the incomplete beta function and the Lerch transcendent. Also, in (22), the integral z 0 tanh α t dt is calculated in terms of elementary functions for α ∈ Q and α > −1. It is worth noting that (22) accelerates quite significantly the symbolic computation of the latter integral with the aid of computer algebra.
Finally, with the aid of the reduction formulas (13) and (16), we have tested that the numerical algorithm provided by MATHEMATICA sometimes fails to compute the imaginary part of B (ν, 0, z). Also, the reduction formulas (13) and (16) are numerically useful to plot B (ν, 0, z) as a function of ν in the real domain.
All the results presented in this paper have been implemented in MATH-EMATICA and can be downloaded from https://bit.ly/2XT7UjK