We outline the systematic calculation of multiple antiderivatives of the Legendre series *q*_{n}(*x*) = *Q*_{n}(1 + 2*x*^{2}), 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 *g*_{k}(*x*)*x*^{l},

- (A3)

where *k*, *l* and *n* are nonnegative integers, and *A*_{k,l,0}(*x*) = *g*_{k}(*x*)*x*^{l}. 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 *A*_{k,l,n}(*x*) in (A3). First,

- (A5)

so that *I* ′ _{1}(*x*,*k*) = *x*2*k*log(1 + *x*2) = : *I*_{0}(*x*,*k*). Odd powers can be dealt with in like manner,

- (A6)

The integrals (A6) are needed to calculate the antiderivative of *I*_{1}(*x*,*k*) in (A5),

- (A7)

The third antiderivative of *I*_{0}(*x*,*k*) reads

- (A8)

where we made use of identity *A*_{k + 1,0,1}(*x*) = *x* − *A*_{k,2,1}(*x*) in (A4).

The antiderivative of *I*_{3}(*x*,*k*) is

- (A9)

where we employed identity *A*_{k + 1,1,1}(*x*) = *x*^{2} / 2 − *A*_{k,3,1}(*x*), see (A4). In these iterative calculations, , we used the elementary integrals

- (A10)

Apparently, *I* ′ _{n + 1}(*x*,*k*) = *I*_{n}(*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 *A*_{k,l,0}(*x*) can be summed in closed form as *g*_{k}(*x*)*x*^{l}, with *g*_{k}(*x*) in (A1).

In addition to the integrals *I*_{n}(*x*,*k*), we need multiple antiderivatives of *L*_{0}(*x*,*k*) = *x*^{2k}log*x*, that is,

- (A11)

with *n* ≥ 1, as well as the antiderivatives of *P*_{0}(*x*,*k*) = *x*^{2k},

- (A12)

The *i*-fold antiderivatives of the Legendre series *q*_{n}(*x*), listed in Section 7 for *i* = 1,2,3,4, are assembled from the integrals *I*_{n}(*x*,*k*), *L*_{n}(*x*,*k*), and *P*_{n}(*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 *A*_{i}(*x*,*n*), we interchange the *k* summation with the *j* summation of *A*_{k,l,n}(*x*) (defined in (A3) and occurring in the series representation of the integrals *I*_{i}(*x*,*k*)),

- (A17)

In the following, we list *A*_{i}(*x*,*n*) for *i* = 1,2,3,4. By making use of the antiderivative *I*_{1}(*x*,*k*) in (A5) and the summation interchange (A17), we find, see (A14),

- (A18)

where the finite series *a*_{1}(*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 *A*_{2}(*x*,*n*) in (A14) as

- (A19)

with series *a*_{2}(*j*,*n*) defined in (7.6). Here, we used *I*_{2}(*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 *A*_{3}(*x*,*n*) in (A14) is found by substituting integral *I*_{3}(*x*,*k*), see (A8), and by performing the interchange (A17),

- (A20)

Series *a*_{3}(*j*,*n*) is defined in (7.11), and the final result for is recorded in (7.10).

Finally, we substitute integral *I*_{4}(*x*,*n*) (calculated in (A9)) into series *A*_{4}(*x*,*n*) (defined in (A14)), and interchange summations as in (A17), to obtain

- (A21)

Series *a*_{4}(*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 *P*_{n}(*x*) in representation (4.4), by performing a variable change *x* = 1 − 2*y*^{2},

- (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 *A*_{i}(*x*,*n*). That is, the coefficients *A*_{i}(*x*,*n*) required in (A13) are calculated by substituting the integrals *I*_{i}(*x*,*k*) into series (A14) and by performing the *k* summation as indicated in (A14). Closed analytic expressions of the integrals *I*_{i}(*x*,*k*) are given in (A5) and (A7), (A8), (A9), where we use the finite series representation (A3) of the antiderivatives *A*_{k,l,n}(*x*) in the numerical evaluation.