A note on some reduction formulas for the incomplete beta function and the Lerch transcedent
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A note on some reduction formulas for theincomplete beta function and the Lerch transcedent
J.L. Gonz´alez-Santander
Received: date / Accepted: date
Abstract
We derive new reduction formulas for the incomplete beta functionand the Lerch transcendent in terms of elementary functions. As an applica-tion, we calculate some new integrals. Also, we use these reduction formulasto test the performance of the algorithms devoted to the numerical evaluationof the incomplete beta function.
Keywords
Incomplete beta function · Lerch transcendent · reductionformulas · numerical evaluation of special functions Mathematics Subject Classification (2010) · The origins of the beta function B ( ν, µ ) go back to Wallis’ attempt of thecalculation of π [11]. For this purpose, he evaluated the integralB ( ν, µ ) = Z t ν − (1 − t ) µ − dt, (1)where ν and µ are integers or µ = 1 and ν is rational. Moreover, Wallissuggested that [1, p. 4] π Z t − / (1 − t ) / dx = 14 lim n →∞ (cid:18) · · · · · n · · · · · (2 n −
1) 1 √ n (cid:19) . This result may have led Euler to consider the integral (1) for ν and µ not neccesarily integers and its relation to the gamma function. In fact, Euler J.L. Gonz´alez-SantanderDepartment of Mathematics. Universidad de Oviedo.Federico Garc´ıa Lorca 18, 33007 Oviedo, Spain.Tel.: +34-985-10-3338E-mail: [email protected] J.L. Gonz´alez-Santander derived the following relation between the beta and gamma functions [1, Eqn.1.1.13]: B ( ν, µ ) = Γ ( ν ) Γ ( µ )Γ ( ν + µ ) . A natural generalization of the beta function is the incomplete beta func-tion, defined as [8, Eqn. 8.17.1]B ( ν, µ, z ) = Z z t ν − (1 − t ) µ − dt, ≤ z ≤ , ν, µ > , where it is straightforward to continue analytically to complex values of ν , µ ,and z .Many applications have been developed over time regarding the B ( ν, µ, z )function. For instance, in statistics it is used extensively as the probabilityintegral of the beta distribution [5, p. 210-275]. Also, it appears in statisticalmechanics for Monte Carlo sampling [6], in the analysis of packings of gran-ular objects [9], and in growth formulas in cosmology [4]. Therefore, in orderto evaluate the B ( ν, µ, z ) function, it is quite interesting to have reductionformulas to simplify its computation, both symbolically and numerically. Forinstance, when µ = m +1 is a positive integer (i.e. m = 0 , , , . . . ), we have thefollowing reduction formula in terms of elementary functions [7, Eqn. 58:4:3]B ( ν, m + 1 , z ) = z ν m X k =0 (cid:18) mk (cid:19) ( − z ) k k + ν . (2)However, when µ = 0, the incomplete beta function is given in terms ofthe Lerch transcendent [7, Eqn. 58:4:4]B ( ν, , z ) = z ν Φ ( z, , ν ) , ν > , (3)where the Lerch transcendent is defined as [2, Eqn. 1.11(1)]Φ ( z, s, ν ) = ∞ X k =0 z k ( k + ν ) s , | z | < , ν = 0 , − , − , . . . (4)It is worth noting that (2) can be proved by induction from (3) and (4),applying the connection formula [7, Eqn. 58:5:3]:B ( ν, µ, z ) = B ( ν + 1 , µ, z ) + B ( ν, µ + 1 , z ) . Nevertheless, reduction formulas for B ( ν, , z ) when ν is a rational numberdo not seem to be reported in the most common literature. The aim of this noteis just to provide such reduction formulas in terms of elementary functions.As an application, we will calculate some new integrals in terms of elementaryfunctions. Also, we will check that the numerical evaluation of the incompletebeta function is improved with these reduction formulas.This paper is organized as follows. Section 2 derives reduction formulas forB ( ν, , z ), both for ν positive rational, as well as negative rational. Particular eduction formulas for incomplete beta and Lerch transcendent 3 cases for ν non-negative integer or ν half-integer are also derived. In Section3, we will apply the reduction formulas derived in Section 2 to calculate someintegrals which do not seem to be reported in the most common literature.Further, for particular values of the parameters, the symbolic computationof these integrals is quite accelerated by using the aforementioned reductionformulas. Also, we will use these reduction formulas to numerically test theperformance of the algorithm provided in MATHEMATICA to compute theincomplete beta function. First, note that according to (3) and (4),B ( ν, , z ) = ∞ X k =0 z k + ν k + ν , (5)thus B ( ν, , z ) is divergent for non-positive integral values of ν . Therefore, wewill consider two different cases in this Section: ν ∈ Q + and ν ∈ Q − \ {− , − , . . . } .2.1 Case ν ∈ Q + Consider ν = n + r > n = ⌊ ν ⌋ ≥ ν and0 ≤ r ≤
1. From (5), we haveB ( n + r, , z ) = ∞ X k =0 z k + n + r k + n + r = ∞ X k = n z k + r k + r = ∞ X k =0 z k + r k + r − n − X k =0 z k + r k + r . (6)Set r = 1 in (6) and then apply the Taylor expansion [8, Eqn. 4.6.1]log (1 + z ) = − ∞ X k =1 ( − z ) k k , to obtain B ( n + 1 , , z ) = − log (1 − z ) − n X k =1 z k k , (7) n = 0 , , , . . . Further, set r = 1 / − z = ∞ X k =0 z k +1 k + 1 , (8) J.L. Gonz´alez-Santander to obtain B (cid:18) n + 12 , , z (cid:19) = 2 tanh − √ z − n − X k =0 z k +1 / k + 1 ! , (9) n = 0 , , , . . . More generally, set r = p/q ∈ Q in (6) with p < q . Then,B (cid:18) n + pq , , z (cid:19) = z p/q ∞ X k =0 z k k + p/q − n − X k =0 z k + p/q k + p/q , (10)Rewrite the first sum of (10) as a hypergeometric function (see [1, p. 61-62]), ∞ X k =0 z k k + p/q = 1 p/q F (cid:18) , p/q p/q (cid:12)(cid:12)(cid:12)(cid:12) z (cid:19) . (11)Apply now the reduction formula [10, Eqn. 7.3.1.131] F (cid:18) , p/q p/q (cid:12)(cid:12)(cid:12)(cid:12) z (cid:19) = − pq z − p/q q − X k =0 exp (cid:18) − πipkq (cid:19) log (cid:18) − z /q exp (cid:18) πikp (cid:19)(cid:19) , (12) p, q = 1 , , . . . ; p ≤ q. Therefore, taking into account (11)-(12), rewrite (10) as the following re-sult.
Theorem 1
For ν = n + pq ∈ Q + , with n = ⌊ ν ⌋ and p < q , the reductionformula B ( ν, , z ) = z ν Φ ( z, , ν ) (13)= − q − X k =0 exp (cid:18) − πipkq (cid:19) log (cid:18) − z /q exp (cid:18) πikq (cid:19)(cid:19) − n − X k =0 z k + p/q k + p/q . holds true.Remark 1 Notice that the reduction formula (9) is included in (13), but not(7), which is a singular case. eduction formulas for incomplete beta and Lerch transcendent 5 ν ∈ Q − \ {− , − , . . . } Consider ν = − n + r < n = ⌊| ν − |⌋ ≥
0, and 0 < r <
1. From (5),we have B ( − n + r, , z ) = ∞ X k =0 z k − n + r k − n + r = ∞ X k = − n z k + r k + r = ∞ X k =0 z k + r k + r + n X k =1 z − k + r − k + r . (14)Taking r = 1 / (cid:18) − n + 12 , , z (cid:19) = 2 tanh − √ z − n X k =1 z − k +1 / k − ! . (15)More generally, take r = p/q ∈ Q with p < q in (14), and apply (11) toobtain the following result. Theorem 2
For ν = − n + pq ∈ Q − , with n = ⌊| ν − |⌋ and p < q , thereduction formula B ( ν, , z ) (16)= − q − X k =0 exp (cid:18) − πipkq (cid:19) log (cid:18) − z /q exp (cid:18) πikq (cid:19)(cid:19) + n X k =1 z p/q − k p/q − k . holds true.Remark 2 Notice that (15) is included in (16) as a particular case. Also, in(16), B ( ν, , z ) = z ν Φ ( z, , ν ) since (3) does not hold true for ν < From the reduction formulas obtained in Section 2, next we calculate someintegrals in terms of elementary functions. Also, we will use these reductionformulas as a benchmark for the computation of the incomplete beta function.3.1 Calculation of integralsOn the one hand, an integral representation of the incomplete beta functionis given by [7, Eqn. 58:3:5],B ( ν, µ, z ) = z ν Z t ν − (1 − zt ) µ − dt. (17) J.L. Gonz´alez-Santander
Also, an integral representation of the Lerch transcendent is [2, Eqn. 1.11(3)]Φ ( z, s, ν ) = 1Γ ( s ) Z ∞ t s − e − ( ν − t e t − z dt, Re ν > . (18)Notice that from (3), (17), and (18), we have Z t ν − − zt dt = Z ∞ e − ( ν − t e t − z dt = z − ν B ( ν, , z ) , (19)Re ν > . Therefore, from (13) and (19), we have for ν = n + pq ∈ Q + , with n = ⌊ ν ⌋ and p < q , Z t ν − − zt dt = Z ∞ e − ( ν − t e t − z dt (20)= − z − n − p/q q − X k =0 exp (cid:18) − πipkq (cid:19) log (cid:18) − z /q exp (cid:18) πikq (cid:19)(cid:19) − n − X k =0 z k − n k + p/q . On the other hand, in the literature we found [7, Eqn. 58:14:7] Z z tanh λ − t dt = 12 B (cid:0) λ, , tanh z (cid:1) , Re λ > . (21)Therefore, from (13) and (21), we have for λ = n + pq ∈ Q + , with n = ⌊ λ ⌋ and p < q , Z z tanh λ − t dt (22)= − q − X k =0 exp (cid:18) − πipkq (cid:19) log (cid:18) − (tanh z ) /q exp (cid:18) πikq (cid:19)(cid:19) − n − X k =0 (tanh z ) k + p/q ) k + p/q . The integral given in (22) generalizes the results found in the literature for λ = n + 1 and λ = n + with n = 0 , , , . . . [3, Eqns. 2.424.2-3].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 = , we obtain Z z tanh / t dt = tanh − (cid:16) √ tanh z (cid:17) − √ tanh z + tan − (cid:16) √ tanh z (cid:17) , but the Integrate command takes around 300 times longer than the reductionformula given in (22). eduction formulas for incomplete beta and Lerch transcendent 7
Fig. 1
Evaluation of Im (B ( ν, , z )) with MATHEMATICA and (13) with ν = 12 . ν, , z ) as a function of ν in the real domain. However, forsome real values of ν and z , we obtain a complex value for B ( ν, , z ). In thesecases, the imaginary part of B ( ν, , z ) is not always easy to compute. Figure 1shows the plot of Im (B ( ν, , z )) as a function of z for ν = 12 .
3. The reductionformula (13) shows the correct answer, i.e. Im (B ( ν, , z )) = − π , meanwhile thenumerical evaluation of Im (B ( ν, , z )) with MATHEMATICA diverges fromthis 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, , ν ))with MATHEMATICA yields also − π . On the one hand, we have derived in (13) and (16) new expressions for theincomplete beta function B ( ν, , z ) and the Lerch transcendent Φ ( z, , ν ) interms of elementary functions when ν is rational and z is complex. Particularformulas 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 thereduction formulas (13) and (16) and the integral representation of the in-complete beta function and the Lerch transcendent. Also, in (22), the integral R z 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 sym-bolic computation of the latter integral with the aid of computer algebra.Finally, with the aid of the reduction formulas (13) and (16), we have testedthat the numerical algorithm provided by MATHEMATICA sometimes failsto compute the imaginary part of B ( ν, , z ). Also, the reduction formulas (13) J.L. Gonz´alez-Santander and (16) are numerically useful to plot B ( ν, , z ) as a function of ν in the realdomain.All the results presented in this paper have been implemented in MATH-EMATICA and can be downloaded from https://bit.ly/2XT7UjK Conflict of interest
The authors declare that they have no conflict of interest.