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Families of Log Legendre Chi Function Integrals

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Families of Log Legendre Chi Function Integrals Applicable Analysis and Discrete Mathematics available online at http://pefmath.etf.rs Appl. Anal. Discrete Math. x (xxxx), xxx{xxx. doi:10.2298/AADMxxxxxxxx FAMILIES OF LOG LEGENDRE CHI FUNCTION INTEGRALS Anthony Sofo In this paper we investigate the representation of integrals involving the prod- uct of the Legendre Chi function, polylogarithm function and log function. We will show that in many cases these integrals take an explicit form involv- ing the Riemann zeta function, the Dirichlet Eta function, Dirichlet lambda function and many other special functions. Some examples illustrating the theorems will be detailed. 1. INTRODUCTION PRELIMINARIES AND NOTATION In this paper we investigate the representations of integrals of the type 1 Z a b m (1) x χp (x) Liq(δx ) ln (x) dx; 0 in terms of special functions such as zeta functions, Dirichlet eta functions, polylog- arithmic functions, beta functions and others. We note that χp (x) is the Legendre- b Chi function (LCF), Liq(δx ) is the polylogarithmic function, a 2 R, a ≥ −2, b 2 R+, p 2 N, q 2 N, δ 2 [−1; 1] n f0g and m 2 N for the set of complex num- bers C, natural numbers N, the set of real numbers R and the set of positive real numbers, R+: The following notation and results will be useful in the subsequent λ sections of this paper. A generalized binomial coefficient µ (λ, µ 2 C) is defined, in terms of the familiar (Euler's) gamma function, by λ Γ(λ + 1) := (λ, µ 2 ); µ Γ(µ + 1) Γ (λ − µ + 1) C 2020 Mathematics Subject Classification. 11M06, 33B15, 65B10 Keywords and Phrases. Legendre Chi Function, Polylogarithm Function, Euler Sum, Dirichlet Lambda Function, Dirichlet Beta Function 1 2 Anthony Sofo which, in the special case when µ = n (n 2 N0 := N [ f0g); yields λ λ λ (λ − 1) ··· (λ − n + 1) (−1)n (−λ) := 1 and := = n (n 2 ); 0 n n! n! N where (λ)ν (λ, ν 2 C) is the Pochhammer symbol defined, also in terms of the Gamma function, by 8 1 (ν = 0; λ 2 n f0g) Γ(λ + ν) < C (λ) := = ν Γ(λ) : λ(λ + 1) ··· (λ + n − 1) (ν = n 2 N; λ 2 C); it being understood conventionally that (0)0 := 1 and assumed tacitly that the Γ- (p) quotient exists. The generalized p−order harmonic numbers, Hn (α; β) are defined as the partial sums of the modified Hurwitz zeta function X 1 ζ (p; α; β) = : (αn + β)p n≥0 The classical Hurwitz zeta function X 1 ζ (p; a) = (n + a)p n≥0 for Re (p) > 1 and by analytic continuation to other values of p 6= 1, where any term of the form (n + a) = 0 is excluded. Therefore n X 1 H(p) (α; β) = n (αj + β)p j=1 (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 X n! X (n − 1)! β (1) = = 2n+1 3 2n+1 1 n≥0 2 n n≥1 2 n and X n! (1 + Hn (2; 1)) X (n − 1)!hn β (2) = = ; 2n+1 3 2n+1 1 n≥0 2 n n≥1 2 n 1 here β (2) is Catalan's constant and hn = H2n − 2 Hn: Two special cases of the Legendre-Chi function are n X (−1) n!x2n+1 χ1 (x) = 2 n+1 3 n≥0 (1 − x ) 2 n Familes of Legendre Chi Function Integrals 3 and n 2n+1 ! X n! (−1) x Hn (2; 1) χ2 (x) = 1 + : 3 2 n+1 n≥0 2 n (1 − x ) The Catalan constant 1 n+1 X (−1) G = β (2) = 2 ≈ 0:91597 n=1 (2n − 1) is a special case of the Dirichlet beta function 1 n+1 X (−1) β (z) = ; for Re (z) > 0(2) (2n − 1)z n=1 1 1 3 = (z−1) − (z−1) ; (−2)2z (z − 1)! 4 4 with functional equation 2 z πz β (1 − z) = sin Γ(z) β (z) π 2 extending the Dirichlet Beta function to the left hand side of the complex plane Re(z) ≤ 0: The Lerch transcendent, 1 X zm Φ(z; t; a) = t m =0 (m + a) is defined for jzj < 1 and Re (a) > 0 and satisfies the recurrence: Φ(z; t; a) = z Φ(z; t; a + 1) + a−t: It is known that the Lerch transcendent extends by analytic continuation to a function Φ (z; t; a) which is defined for all complex t, z 2 C − [1; 1) and a > 0, which can be represented, [15], by the integral formula 1 1 1 Z xt−1e−(t−1)x 1 Z xa−1 ln 1 Φ(z; t; a) = dx = x dx Γ(t) ex − z Γ(t) 1 − xz 0 0 for Re (t) > 0: The Lerch transcendent generalizes the Hurwitz zeta function at z = 1; 1 X 1 Φ (1; t; a) = ζ (t; a) = t m =0 (m + a) and the polylogarithm, or de-Jonqui`ere'sfunction, when a = 1; 1 X zm Li (z) := ; t 2 when jzj < 1; Re (t) > 1 when jzj = 1: t mt C m =1 4 Anthony Sofo The polylogarithm of negative integer order arises in the sums of the form n−1 X 1 X n jnzj = Li (z) = zn−i −n n+1 i j≥1 (1 − z) i =0 n n + 1 where the Eulerian number = Pi+1 (−1)j (i − j + 1)n. The i j =0 j Legendre-Chi function is a special case of the Lerch transcendent 1 1 X x2n+1 χ (x) = 2−p x Φ x2; p; = p 2 (2n + 1)p n =0 and is related to the polylogarithm by 1 χ (x) = (Li (x) − Li (−x)) = Li (x) − 2−pLi (x2): p 2 p p p p There are many special values of the LCF, from Lewin [19] x 1 Z 1 + t dt χ (x) = ln 2 2 1 − t t 0 and 1 − x 3 1 1 + x χ (x) + χ (x) = ζ (2) + ln x ln ; 2 1 + x 2 4 2 1 − x hence, p 1 3 χ 5 − 2 = ζ (2) − ln2 (φ) 2 4 4 p 1 where the golden ratio φ = 2 1 + 5 : The Dirichlet lambda function, λ (p) is (3) 2λ (p) = 2χp (1) = ζ (p) + η (p) where 1 n+1 X (−1) η (p) = = 1 − 21−p ζ (p) np n =1 is the alternating zeta function and η (1) = ln 2: In the case of the summation of harmonic numbers, we know that the famous Euler identity states, for m 2 N ≥ 2; 1 m−2 X Hn 1 1 X (4) EU (m) = = (m + 2)ζ (m + 1) − ζ (m − j) ζ (j + 1) ; nm 2 2 n =1 j=1 for odd powers of the denominator, Georghiou and Philippou [14] established the identity: 1 2m X Hn 1 X = (−1)r ζ(r) ζ(2m + 2 − r); m ≥ 1: n2m+1 2 n=1 r=2 Familes of Legendre Chi Function Integrals 5 We know that for n ≥ 1; (n + 1) − (1) = Hn with (1) = −γ; where γ is the Euler Mascheroni constant and (n) is the digamma function. For real values of x; (x) is the digamma (or psi) function defined by d Γ0(x) (x) := flog Γ(x)g = ; dx Γ(x) 1 1 1 X 1 1 X 1 1 (x) = −γ − + − = −γ + − ; x n x + n n x + n − 1 n=1 n=1 leading to the telescoping sum: 1 X 1 1 1 (1 + x) − (x) = − = : x + n − 1 x + n x n=1 The polygamma function k 1 d k+1 X 1 (k)(z) = f (z)g = (−1) k! dzk k+1 r=0 (r + z) and has the recurrence (−1)k k! (k)(z + 1) = (k)(z) + : zk+1 The connection of the polygamma function with harmonic numbers is, (−1)m (5) H(m+1) = ζ (m + 1) + (m) (z + 1) ; z 6= {−1; −2; −3; :::g : z m! 1 (−1)m Z (1 − tz) = lnm t dt m! 1 − t 0 and the multiplication formula is p−1 1 X j (6) (k)(pz) = δ ln p + (k)(z + ) p;0 pk+1 p j=0 + (p) for p a positive integer and δp;k is the Kronecker delta. Let a 2 R and Hn denote (p) P1 Han harmonic numbers of order p; and put BW (a; p:q) = n =1 nq : We have, from 6 Anthony Sofo the work of [4] that for odd weight (p + q) 1 (p) X Hn BW (1; p:q) = nq n =1 p [ 2 ] p X p + q − 2j − 1 = (−1) ζ (p + q − 2j) ζ (2j) p − 1 j =1 1 + 1 + (−1)p+1 ζ (p) ζ (q) 2 p [ 2 ] p X p + q − 2j − 1 + (−1) ζ (p + q − 2j) ζ (2j)(7) q − 1 j =1 ζ (p + q) p + q − 1 p + q − 1 + 1 + (−1)p+1 + (−1)p+1 ; 2 p q where [z] is the integer part of z: Let us denote 1 n+1 (p) X (−1) Han S (a; p; q) := ; nq n =1 then in the case where p and q are both positive integers and p+q is an odd integer, Flajolet and Salvy [12] gave the identity: p p X p + i − 1 2S (1; p; q) = (1 − (−1) ) ζ (p) η (q) + 2 (−1) ζ (p + i) η (2k) p − 1 i+2k=q X q + j − 1 j (8) + η (p + q) − 2 (−1) η (q + j) η (2k) ; q − 1 j+2k=p 1 1 where η (0) = 2 ; η (1) = ln 2; ζ (1) = 0; and ζ (0) = − 2 in accordance with the analytic continuation of the Riemann zeta function.
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