Integrals involving the Legendre Chi function
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Integrals involving the Legendre Chi function A. Sofo1 Received: 7 July 2020 / Accepted: 6 November 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract In this paper we investigate the representation of integrals involving the Legendre Chi function. We will show that in many cases these integrals take an explicit form involving the Riemann zeta function, the Dirichlet Eta function, Dirichlet lambda function and many other special functions. Some examples illustrating the theorems will be detailed. Keywords Legendre Chi function · Polylogarithm function · Euler sums · Dirichlet lambda function · Zeta functions · Dirichlet beta functions Mathematics Subject Classification 11M06 · 11M35 · 11Y60 · 33B15 · 41A58 · 65B10
1 Introduction preliminaries and notation In this paper we investigate the representations of integrals of the type 1 χ p (x) f (x) d x,
(1.1)
0
in terms of special functions such as Zeta functions, Dirichlet Eta functions, Polylogarithmic functions, Beta functions and others. In its most general form f (x) = x a χq (x) lnm (x) , x a Liq (δx b ) where χq (x) is the Legendre-Chi function (LCF), Liq (δx b ) is the polylogarithmic function, a ∈ R ≥ −1, b ∈ R+ , p ∈ N, q ∈ N, m ∈ N and δ ∈ [−1, 1], for the set of natural numbers N, the set of real numbers R and the set of positive real numbers, R+ . We shall also investigate some representations of integrals involving the product of LCFs. The following notation and results will be useful in the subsequent sections of this paper. The generalized p( p) order harmonic numbers, Hn (α, β) are defined as the partial sums of the modified Hurwitz
B 1
A. Sofo [email protected] College of Engineering and Science, Victoria University, Melbourne, Australia 0123456789().: V,-vol
123
24
Page 2 of 21
A. Sofo
zeta function
ζ ( p, α, β) =
1 . (αn + β) p
n≥0
The classical Hurwitz zeta function ζ ( p, a) =
n≥0
1 (n + a) p
for Re ( p) > 1 and by analytic continuation to other values of p = 1, where any term of the form (n + a) = 0 is excluded. Therefore ( p)
Hn
(α, β) =
n j=1
1 (α j + β) p ( p)
( p)
and the “ordinary” p-order harmonic numbers Hn = Hn (1, 0) . Many functions can be expanded through the generalized p-order harmonic numbers, such as the Dirichlet Beta cases 1 1 β (1) = = 1 n+ 2 n − 21 n≥0 2n+1 n≥1 2n 1 1 2
2
and β (2) =
1 + Hn (2, 1) hn = 1 n + n − 2 n≥0 2n+1 n≥1 2n 1 1 2
1 2
,
2
here β (2) is Catalan’s constant and h n = function are x(−x 2 )n χ1 (x) = n+ n≥0 (1 − x 2 )n+1 1
H2n − 21 Hn . Two special cases of the Legendre-Chi
1 2
2
and χ2 (x) =
n≥0
1 (
n+ 1 2
1 2
)
n x −x 2 Hn (2, 1) 1+ . (1 − x 2 )n+1
The Catalan constant G = β (2) =
∞ (−1)n+1 ≈ 0.91597 (2n − 1)2 n=1
is a special case of the Dirichlet Beta function ∞ (−1)n+1 , for Re(z) > 0 (2n − 1)z n=1 1 (z−1) 1 (z−1) 3 ψ = − ψ , 4 4 (−2)2z (z − 1)!
β (z) =
123
(1.2)
Integrals involving the Legendre Chi function
with functional equation β (1 − z) =
Page 3 of 21
π z
24
πz
(z
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