We outline the systematic calculation of multiple antiderivatives of the Legendre series qn(x) = Qn(1 + 2x2), see (4.3) and (4.9), as defined in (7.1). In particular, we derive the antiderivatives for k = 1,2,3,4, which are stated in (7.2), (7.5), (7.10), and (7.16).
We start with some prerequisites relating to the geometric series
- (A1)
and its n-fold antiderivative
- (A2)
where we use the rising factorial (a)n = a(a + 1) ⋯ (a + n − 1) and (a)0 = 1. More generally, we will need the antiderivatives of gk(x)xl,
- (A3)
where k, l and n are nonnegative integers, and Ak,l,0(x) = gk(x)xl. We also note the identities
- (A4)
To assemble the antiderivatives of the Legendre series (4.9), we need some integrals that can be expressed in terms of the finite series Ak,l,n(x) in (A3). First,
- (A5)
so that I ′ 1(x,k) = x2klog(1 + x2) = : I0(x,k). Odd powers can be dealt with in like manner,
- (A6)
The integrals (A6) are needed to calculate the antiderivative of I1(x,k) in (A5),
- (A7)
The third antiderivative of I0(x,k) reads
- (A8)
where we made use of identity Ak + 1,0,1(x) = x − Ak,2,1(x) in (A4).
The antiderivative of I3(x,k) is
- (A9)
where we employed identity Ak + 1,1,1(x) = x2 / 2 − Ak,3,1(x), see (A4). In these iterative calculations, , we used the elementary integrals
- (A10)
Apparently, I ′ n + 1(x,k) = In(x,k), which serves as a consistency check of the antiderivatives (A5) and (A7), (A8), (A9). When differentiating, we employ the differential identity in (A4) and make use of the fact that series Ak,l,0(x) can be summed in closed form as gk(x)xl, with gk(x) in (A1).
In addition to the integrals In(x,k), we need multiple antiderivatives of L0(x,k) = x2klogx, that is,
- (A11)
with n ≥ 1, as well as the antiderivatives of P0(x,k) = x2k,
- (A12)
The i-fold antiderivatives of the Legendre series qn(x), listed in Section 7 for i = 1,2,3,4, are assembled from the integrals In(x,k), Ln(x,k), and Pn(x,k), calculated in (A5), (A6), (A7), (A8), (A9), (A10), (A11), (A12). Substituting the Legendre series (4.9) into the multiple integral (7.1), we find
- (A13)
where
- (A14)
- (A15)
- (A16)
In series Ai(x,n), we interchange the k summation with the j summation of Ak,l,n(x) (defined in (A3) and occurring in the series representation of the integrals Ii(x,k)),
- (A17)
In the following, we list Ai(x,n) for i = 1,2,3,4. By making use of the antiderivative I1(x,k) in (A5) and the summation interchange (A17), we find, see (A14),
- (A18)
where the finite series a1(j,n) is stated in (7.3). The antiderivative recorded in (7.2) is obtained by means of (A13) with (A15), (A16), and (A18) substituted.
As for the second antiderivative , see (A13), we obtain A2(x,n) in (A14) as
- (A19)
with series a2(j,n) defined in (7.6). Here, we used I2(x,k) in (A7) and interchanged summations according to the prescription (A17). The final result for is stated in (7.5).
Regarding the calculation of in (A.13), we can proceed analogously. Series A3(x,n) in (A14) is found by substituting integral I3(x,k), see (A8), and by performing the interchange (A17),
- (A20)
Series a3(j,n) is defined in (7.11), and the final result for is recorded in (7.10).
Finally, we substitute integral I4(x,n) (calculated in (A9)) into series A4(x,n) (defined in (A14)), and interchange summations as in (A17), to obtain
- (A21)
Series a4(j,n) is defined in (7.17). The fourth antiderivative of the Legendre series (4.9) is assembled via (A13) by substitution of (A15), (A16), and (A21). The final result is stated in (7.16).
In (A18), (A19), (A20), (A21), we encounter series of type
- (A22)
where Reα > 0, and the Hankel symbol [n,k] is defined in (2.4). All finite series arising in (A18), (A19), (A20), (A21), which do not involve powers of x, can be reduced to S(n,α) by partial fraction decomposition. Series (A22) can be traced back to the Legendre polynomial Pn(x) in representation (4.4), by performing a variable change x = 1 − 2y2,
- (A23)
Thus, we may write series (A22) as
- (A24)
By making use of (A23) and a standard integral of the hypergeometric function [13], we find
- (A25)
where n ≥ 0 is a nonnegative integer, and Reα > 0. In this way, we have summed series (A22). For instance,
- (A26)
Singularities emerge in the gamma functions in (A25) at positive integer α. In this case, epsilon expansion is needed. We put α = m + ε, m ≥ 1, and use the expansion
- (A27)
valid for integer j ≥ 0, where the psi function is defined in (4.7). We find S(n,m + ε) = O(ε) for n ≥ m and
- (A28)
valid for n < m. Thus,
- (A29)
When assembling the antiderivatives in Section 7, we make extensive use of series S(n,α) in (A22) for half-integer and integer α as listed in (A26) and (A29).
Finally we note a consistency check of the antiderivatives in (7.2), (7.5), (7.10), and (7.16). The numerical calculation of can directly be based on (A13), (A14), (A15), (A16), without the interchange of indices (A17) in Ai(x,n). That is, the coefficients Ai(x,n) required in (A13) are calculated by substituting the integrals Ii(x,k) into series (A14) and by performing the k summation as indicated in (A14). Closed analytic expressions of the integrals Ii(x,k) are given in (A5) and (A7), (A8), (A9), where we use the finite series representation (A3) of the antiderivatives Ak,l,n(x) in the numerical evaluation.