The dilogarithm integral Li(xs) and its associated functions Li+(xs) and Li-(xs) are defined as locally summable functions on the real line. Some convolutions and neutrix convolutions of these functions and other functions are then found.

1. Introduction

The dilogarithm integral Li(x) is defined byLi(x)=-∫0xln|1-t|tdt
(see [1]). More generally, we have Li(xs)=-∫0xsln|1-t|tdt=-s∫0xln|1-ts|tdt
for s=1,2,….

The associated functions Li+(xs) and Li-(xs) are defined by Li+(xs)=H(x)Li(xs),Li-(xs)=H(-x)Li(xs)=Li(xs)-Li+(xs),
where H(x) denotes Heaviside's function.

Next, we define the distribution ln|1-xs|x-1 by ln|1-xs|x-1=-s-1[Li(xs)]′,
and its associated distributions ln|1-xs|x+-1 and ln|1-xs|x--1 are defined by ln|1-xs|x+-1=H(x)ln|1-xs|x-1=-s-1[Li+(xs)]′,ln|1-xs|x--1=H(-x)ln|1-xs|x-1=-s-1[Li-(x)]′.

The classical definition of the convolution product of two functions f and g is as follows.

Definition 1.1.

Let f and g be functions. Then the convolution f*g is defined by (f*g)(x)=∫-∞∞f(t)g(x-t)dt
for all points x for which the integral exist.

It follows easily from the definition that if f*g exists then g*f exists and f*g=g*f,
and if (f*g)′ and f*g′(orf′*g) exists, then (f*g)′=f*g′(orf′*g).
Definition 1.1 can be extended to define the convolution f*g of two distributions f and g in 𝒟′ with the following definition; see Gel'fand and Shilov [2].

Definition 1.2.

Let f and g be distributions in 𝒟′. Then the convolution f*g is defined by the equation 〈(f*g)(x),φ〉=〈f(y),〈g(x),φ(x+y)〉〉
for arbitrary φ in 𝒟, provided f and g satisfy either of the following conditions:

either f or g has bounded support,

the supports of f and g are bounded on the same side.

It follows that if the convolution f*g exists by this definition then (1.7) and (1.8) are satisfied.

In order to extend Definition 1.2 to distributions which do not satisfy conditions (a) or (b), let τ be a function in 𝒟, see [3], satisfying the conditions:

(i) τ(x)=τ(-x),

(ii) 0≤τ(x)≤1,

(iii) τ(x)=1for|x|≤1/2,

(iv) τ(x)=0for|x|≥1.

The function τn is then defined by τn(x)={1,|x|≤n,τ(nnx-nn+1),x>n,τ(nnx+nn+1),x<-n,
for n=1,2,….

The following definition of the noncommutative neutrix convolution was given in [4].

Definition 1.3.

Let f and g be distributions in 𝒟′, and let fn=fτn for n=1,2,…. Then the noncommutative neutrix convolution f⊛g is defined as the neutrix limit of the sequence {fn*g}, provided the limit h exists in the sense that N-limn→∞〈fn*g,φ〉=〈h,φ〉
for all φ in 𝒟, where N is the neutrix, see van der Corput [5], having domain N′ the positive reals and range N′′ the real numbers, with negligible functions finite linear sums of the functions
nλlnr-1n,lnrn:λ>0,r=1,2,…
and all functions which converge to zero in the normal sense as n tends to infinity.

In particular, if
limn→∞〈fn*g,φ〉=〈h,φ〉
exists, we say that the non-commutative convolution f⊛g exists.

It is easily seen that any results proved with the original definition of the convolution hold with the new definition of the neutrix convolution. Note also that because of the lack of symmetry in the definition of f⊛g the neutrix convolution is in general non-commutative.

The following results proved in [4] hold, first showing that the neutrix convolution is a generalization of the convolution.

Theorem 1.4.

Let f and g be distributions in 𝒟′, satisfying either condition (a) or condition (b) of Gel'fand and Shilov's definition. Then the neutrix convolution f⊛g exists and
f⊛g=f*g.

Theorem 1.5.

Let f and g be distributions in 𝒟′ and suppose that the neutrix convolution f⊛g exists. Then the neutrix convolution f⊛g′ exists and
(f⊛g)′=f⊛g′.

Note however that (f⊛g)′ is not necessarily equal to f′⊛g but we do have the following theorem.

Theorem 1.6.

Let f and g be distributions in 𝒟′ and suppose that the neutrix convolution f⊛g exists. If N-limn→∞〈(fτn′)*g,φ〉 exists and equals 〈h,φ〉 for all φ in 𝒟, then f′⊛g exists and
(f⊛g)′=f′⊛g+h.

2. Main Result

We define the function Is,r(x) by Is,r(x)=∫0xurln|1-us|du
for r=0,1,2,… and s=1,2,…. In particular, we define the function Ir(x) by Ir(x)=I1,r(x)
for r=0,1,2,….

The following theorem was proved in [6].

Theorem 2.1.

The convolutions Li+(x)*x+randln|1-x|x+-1*x+rexistandLi+(x)*x+r=1r+1∑i=0r(r+1i)(-1)r-iIr-i(x)x+i+1r+1x+r+1Li+(x)
for r=0,1,2,… and
ln|1-x|x+-1*x+r=∑i=0r-1(ri)(-1)r-iIr-i-1(x)x+i-Li+(x)x+r
for r=1,2,….

We now prove the following generalization of Theorem 2.1.

Theorem 2.2.

The convolutions Li+(xs)*x+r and ln|1-xs|x+-1*x+r exist and
Li+(xs)*x+r=sr+1∑i=0r(r+1i)(-1)r-iIs,r-i(x)x+i+1r+1Li+(xs)x+r+1
for r=0,1,2,…,s=1,2,… and
ln|1-xs|x+-1*x+r=∑i=0r-1(ri)(-1)r-iIs,r-i-1(x)x+i-1sLi+(xs)x+r
for r,s=1,2,….

Proof.

It is obvious that Li+(xs)*x+r=0 if x<0.

When x>0, we have
Li+(xs)*x+r=-s∫0x(x-t)r∫0tu-1ln|1-us|dudt=-s∫0xu-1ln|1-us|∫ux(x-t)rdtdu=sr+1∑i=0r+1(-1)r-ixi(r+1i)∫0xur-iln|1-us|du=sr+1∑i=0r(r+1i)(-1)r-ixiIs,r-i(x)+1r+1xr+1Li(xs)
proving (2.5).

Next, using (1.8) and (2.5), we have
-sln|1-xs|x+-1*x+r=rLi+(xs)*x+r-1=s∑i=0r-1(ri)(-1)r-i-1Is,r-i-1(x)x+i+Li+(xs)xr,
and (2.6) follows.

Corollary 2.3.

The convolutions Li-(xs)*x-r and ln|1-xs|x--1*x-r exist and
Li-(xs)*x-r=sr+1∑i=0r(r+1i)(-1)r-i+1Is,r-i(x)x-i+1r+1Li-(xs)x-r+1
for r=0,1,2,…,s=1,2,… and
ln|1-xs|x--1*x-r=∑i=0r-1(ri)(-1)r-i+1Is,r-i-1(x)x-i-1sLi-(xs)x-r
for r,s=1,2,….

Proof.

Equations (2.9) and (2.10) are obtained applying a similar procedure as used in obtaining (2.5) and (2.6).

The next two theorems were proved in [6] and to prove it, our set of negligible functions was extended to include finite linear sums of the functions nsLi(nr) for s=0,1,2,… and r=1,2,….

Theorem 2.4.

The convolution Li+(x)⊛xr exists and
Li+(x)⊛xr=1r+1∑i=0r(r+1i)(-1)r-i(r-i+1)2xi
for r=0,1,2,….

Theorem 2.5.

The convolution ln|1-x|x+-1⊛xr exists and
ln|1-x|x+-1⊛xr=∑i=0r-1(ri)(-1)r-i+1(r-i)2xi
for r=1,2,….

Before proving some further results, we need the following lemma.

Lemma 2.6.

If r+1/s∈N for r,s=1,2,…, then
N-limn→∞Is,r(n)=-s(r+1)2.

Proof.

Because
Is,r(n)=1r+1(nr+1-1)ln|1-ns|-1r+1∫0ns1-t(r+1)/s1-tdt
when r+1/s∈N, we have
Is,r(n)=1r+1(nr+1-1)ln|1-ns|-1r+1∑i=0(r+1)/s-1nsi+si+1=1r+1(nr+1-1)(slnn+ln|1-n-s|)-1r+1∑i=0(r+1)/s-1nsi+si+1,
and (2.13) follows.

We now prove the following generalization of Theorems 2.4 and 2.5.

Theorem 2.7.

The neutrix convolution Li+(xs)⊛xr exists when r+1/s∈N and
Li+(xs)⊛xr=s2r+1∑i=0r(r+1i)(-1)r-i(r-i+1)2x+i
for r,s=1,2,….

Proof.

We put [Li+(xs)]n=Li+(xs)τn(x). Then the convolution [Li+(xs)]n*xr exists by Definition 1.1 and
[Li+(xs)]n*xr=∫0nLi(ts)(x-t)rdt+∫nn+n-nτn(t)Li(ts)(x-t)rdt=I1+I2,
where
I1=∫0nLi(ts)(x-t)rdt=-s∫0n(x-t)r∫0tln|1-us|ududt=-s∫0nln|1-us|u∫un(x-t)rdtdu=sr+1∑i=0r+1(-1)r-ixi(r+1i)∫0nu-1ln|1-us|{ur-i+1-nr-i+1}du=sr+1∑i=0r(r+1i)(-1)r-ixiIs,r-i(n)+1r+1xr+1Li(ns)+1r+1∑i=0r(r+1i)(-1)r-ixiLi(ns)nr-i+1.
Thus, using Lemma 2.6, we have
N-limn→∞I1=s2r+1∑i=0r(r+1i)(-1)r-i+1(r-i+1)2xi.

Further, it is easily seen that
limn→∞∫nn+n-nτn(t)Li(ts)(x-t)rdt=0,
and (2.16) follows from (2.17), (2.19), and (2.20), proving the theorem.

Theorem 2.8.

The neutrix convolution ln|1-xs|x+-1⊛xr exists when r+1/s∈N and
ln|1-xs|x+-1⊛xr=s∑i=0r-1(ri)(-1)r-i(r-i)2x+i
for r,s=1,2,….

Proof.

Using Theorems 1.5 and 1.6, we have
-sln|1-xs|x+-1⊛xr=rLi+(xs)⊛xr-1+N-limn→∞[Li+(xs)τn′(x)]*xr,
where, on integration by parts, we have
[Li+(xs)τn′(x)]*xr=∫nn+n-nτn′(t)Li(ts)(x-t)rdt=-Li(ns)(x-n)r-s∫nn+n-nln|1-ts|t-1(x-t)rτn(t)dt+r∫nn+n-nLi(ts)(x-t)r-1τn(t)dt.

It is clear that
limn→∞∫nn+n-nln|1-ts|t-1(x-t)rτn(t)dt=0limn→∞∫nn+n-nLi(ts)(x-t)r-1τn(t)dt=0.
It now follows from (2.23) and (2.24) that
N-limn→∞[Li+(xs)τn′(x)]*xr=0.
Equation (2.21) now follows directly from (2.16) and (2.22), proving the theorem.

Corollary 2.9.

The neutrix convolutions Li-(xs)⊛xr and ln|1-xs|x--1⊛xr exist when r+1/s∈N and
Li-(xs)⊛xr=s2r+1∑i=0r(r+1i)(-1)r-i+1(r-i+1)2x-i,ln|1-xs|x--1⊛xr=s∑i=0r-1(ri)(-1)r-i+1(r-i)2x-i
for r,s=1,2,…

Proof.

Equations (2.26) are obtained applying a similar procedure as used in obtaining (2.16) and (2.21).

Corollary 2.10.

The neutrix convolutions Li+(xs)⊛x-r and Li-(xs)⊛x+r exist when r+1/s∈N and
Li+(xs)⊛x-r=sr+1∑i=0r(r+1i)(-1)i(r-i+1)2[s-(r-i+1)2Is,r-i(x)]x+i+(-1)r+1r+1Li+(xs)xr+1,Li-(xs)⊛x+r=sr+1∑i=0r(r+1i)(-1)i+1(r-i+1)2[s-(r-i+1)2Is,r-i(x)]x-i+(-1)r+1r+1Li-(xs)xr+1
for r,s=1,2,….

Proof.

Since the neutrix convolution product is distributive with respect to addition, we have
Li+(xs)⊛xr=Li+(xs)*x+r+(-1)rLi+(xs)⊛x-r,
and (2.27) follows from (2.16) and (2.5). Equation (27) is obtained applying similar procedure as in the case of (2.27).

Corollary 2.11.

The neutrix convolutions ln|1-xs|x+-1⊛x-r and ln|1-xs|x--1⊛x+r exist when r+1/s∈N and
ln|1-xs|x+-1⊛x-r=∑i=0r-1(ri)(-1)i(r-i)2[s-(r-i)2Is,r-i-1(x)]x+i+(-1)rsLi+(xs)xr,ln|1-xs|x--1⊛x+r=∑i=0r-1(ri)(-1)i+1(r-i)2[s-(r-i)2Is,r-i-1(x)]x-i+(-1)rsLi-(xs)xr
for r,s=1,2,…

Proof.

Equation (2.29) follows from (2.21) and (2.6). Equation (29) is obtained applying similar procedure as in the case of (2.29).

Acknowledgment

This research was supported by FEIT, University of Ss. Cyril and Methodius in Skopje, Republic of Macedonia, Project no. 08-3619/8.

AbramowitzM.StegunI. A.Gel'fandI. M.ShilovG. E.JonesD. S.The convolution of generalized functionsFisherB.Neutrices and the convolution of distributionsvan der CorputJ. G.Introduction to the neutrix calculusJolevska-TuneskaB.FisherB.ÖzçağE.On the dilogarithm integral