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11

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differentiation and logarithm

Abstract. Let log¡dxd¢ be the generator of the1-parameter group

{dxdaa|a ∈ R} of fractional order differentiations acting on the space of

operators of Mikusinski ([5]). The Lie algebraglog generated by log¡dx

and logxis a deformation and can be regarded as the logarithm of Heisen-berg Lie algebra. We showglog is isomorphic to the Lie algebra generated

by d

dslog(Γ(1 +s)) and d

ds. Hence as a module, glog is isomorphic to

the module generated by d

ds and polygamma functions. Structure of the

group generated by 1-parameter groups {dxdaa|a∈R} and {x

a|a R}, is

also determined.

M.S.C. 2010: 17B65, 26A33, 44A15, 81R10.

Key words: logarithm of differentiation; integral transformation; deformation of Heisenberg Lie algebra; polygamma function.

Introduction

Schr¨odinger representation of Heisenberg Lie algebra is generated by dxd andx. Re-placing dxd byF(dxd),

F(X) =X

n

cnXn, F

µ

d dx

=X

n

cn

dn

dxn,

we obtain a deformation of Heisenberg Lie algebra. This algebra is isomorphic to the Lie algebra generated by dxd and F(x). It is nilpotent if F(x) is a polynomial and generalized nilpotent ifF(x) is an infinite series..

An example of such deformation is the algebra generated by logarithm of differ-entiation log¡d

dx

¢

and logx.

If we consider fractional order differentiation dxdaa acts on the space of Operators

of Mikusinski ([5]), we can define log¡dxd¢by lima→0dad da

dxa|a=0. Explicitly, we have

log µ d

dx

f(x) =−

µ

γf(x) + Z x

0

log(x−t)df+

dt dt

,

Balkan Journal of Geometry and Its Applications, Vol.16, No.1, 2011, pp. 1-11. c

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whereγ is the Euler constant and df+

dt = df

dt +f(0)δ, δ is the Dirac function ([4, §2

Prop.1.], [6]). By using logarithm of differentiation and the formula

(1.1) d

a

dxax

c= Γ(1 +c)

Γ(1 +c−a)x

c−a,

wherecandc−aare both not negative integers, we obtain the following arguments on fractional calculus:

LetRbe an integral transformation from functions onRto functions on positive

real axis defined by

(1.2) R[f(s)](x) =

Z ∞ −∞

xs f(s)

Γ(1 +s)ds.

Then we have

da

dxaR[f(s)](x) =R[τaf(s)](x), τaf(s) =f(s+a),

(1.3)

log µ d

dx

R[f(s)](x) =R

·df(s)

ds

¸ (x) (1.4)

(§3, Theorem 3.1 and its Corollary). Our study on the structures of glog and the

group generated by exponential image ofglog are based on these equalities.

By the variable changex=et, we havexs=ets. Hence we have

R[f(s)](x) =L

· f(s) Γ(1 +s)

¸

(t), L[g(s)](t) = Z ∞

−∞

estg(s)ds.

Therefore we obtain

da

dxa

¯ ¯ ¯ ¯x=et

L[f(s)](t) =L

·

τa

µ

Γ(1 +s) Γ(1 +s−a)f(s)

¶¸ (t),

(1.5)

log µ

d dx

¶¯ ¯ ¯ ¯

x=et

L[f(s)](t) =L

·µ

d ds+

Γ′(1 +s)

Γ(1 +s) ¶

f(s) ¸

(t).

(1.6)

By (1.6),glogis isomorphic to the Lie algebra generated by dtd andΓ

(1+s)

Γ(1+s) (§4.Theorem

4.2). As a module, this algebra is generated by d dt and ψ

(m)(1 +s), m= 0,1,2, . . ..

Here ψ(m)(s) is the m-th polygamma function dm

dtmψ(s), ψ(s) =ψ(0)(s) =

Γ′

(s) Γ(s) ([1,

§6.4]).

Since ealog(dxd) = d a

dxa and eat =xa, and eat acts as the translation operator τa

in the images of Laplace transformation, the group Glog generated by 1-parameter

groups{dxdaa|a∈R} and{xa|a∈R} is the crossed product Glog ∼=R ⋉GΓ, where

GΓ is the group generated by{Γ(1+Γ(1+s+s)a)|a∈R} by multiplication ([3, §5. Prop.2.]).

Definition ofGΓ in [3] is different. But it gives same group).

Glog is the essential part of the groupGΨ generated by exponential images of the

elements ofglog. Precise structures ofGΨ and generalization of this construction to

the Heisenberg Lie algebra generated byx1, . . . , xn and ∂x1, . . . ,∂xn∂ are also studied

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In Appendix, we give an alternative proof of Theorem 3.1, which derives the integral transformationRnaturally.

Note. Results in§4 and§5 are improvements of our previous results given in [4] and [3], while results in§3 and§6 are new.

differentiation

Letf(x) be a function on positive real axis, and leta >0. Thena-th order indefinite integral off from the origin is given by the Riemann-Liouville integral

(2.1) Iaf(x) = 1

Γ(a) Z x

0

(x−t)a−1f(t)dt.

Hence we may define (n−a)-th order differentiationdxdnn−−aa off by dn

dxnIaf

(Riemann-Liouville) orIa(dnf

dxn) (Caputo). They are different if we consider in the category of

functions. But if we use the space of operators of Mikusinski ([5]), they coincide and {dxdaa|a ∈ R} {

da

dxa|a ∈ R} becomes a 1-parameter group. As a price, we can

not investigate fractional order functions. The constant function 1 is replaced by the Heaviside functionY. Its derivative is the Dirac functionδ.

Proposition 2.1. The generating operator log¡d dx

¢ = d

da da

dxa|a=0 of the 1-parameter group{dxdaa|a∈R} is given by

log µ

d dx

¶ =−

µ

γf(x) + Z x

0

log(x−t)df+

dt dt

,

(2.2)

=−(logx+γ)f(x)−

Z x

0

log µ

1− t

x

df+

dt dt.

(2.3)

Hereγ is the Euler constant and df+

dx means df

dx+f(0)δ ([4, 6]).

Note. If we assumef(0) = 0, or replacef(x) be f(x)−f(0), then we can avoid the use of distribution (cf.[2]).

By definition, we have

(2.4) ealog(dxd) = d

a

dxa.

We also have

log µ

d

dx+g(x)

=G(x)−1log µ

d dx

·G(x), G(x) =eR0xg(t)dt.

Because we have dxd +g(x) = G(x)−1d

dx ·G(x), where G(x) is regarded as a linear

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Example. By (2.2), we have

log µ d

dx

xc=

Ã

logx+ Ã

γ−

X

n=1

c n(n+c)

!!

xc,

log µ

d dx

xn=−

µ

logx+γ−

µ 1 +1

2 +· · ·+ 1

n

¶¶

xn.

We also have

log µ

d dx

(logx)n=−(logx+γ)(logx)n+

+

n−1

X

k=0

(−1)n−k−1n!ζ(nk+ 1)

k ! (logx)

k.

Here ζ(k) is the value of Riemann’s ζ-function at k. Introducing an infinite order differential operatordlog by

(2.5) dlog,X =

d

dtlog(Γ(1 +t))

¯ ¯ ¯ ¯

t= d dX

= Ã

−γ+

X

n=1

(−1)n−1ζ(n+ 1) d

n

dXn

!

,

we have log(d/dx)(logx)n = (X+d

log,X)Xn|X=logx.

calculus

We introduce an integral transformationRby

(3.1) R[f(s)](x) = Z ∞

−∞

xs f(s)

Γ(1 +s)ds, x >0.

To defineR[f], f needs to satisfy some estimate. For example, iff(s) satisfies

(3.2) |f(s)|=O(eM s), s→ ∞, |f(s)|=O(e−|s|α), s→ −∞, α >1,

thenR[f] is defined. But appropriate domain and range ofRare not known.

Note. In this paper, we consider R[f](x) to be a function on positive real axis.But it is better to consider R[f](x) to be a (many valued) function on C× = C\ {0}.

ThenR[f](eit) is defined as a function onR. Here we need to considerR[f](eit) and R[f](ei(t+2π)) take different values.. Then by Fourier inversion formula, we have the

following inversion formula

(3.3) f(s) = Γ(1 +s)

Z ∞ −∞

e−itsR[f](eit)dt.

This shows ifR[f](x) is a periodic function on the unit circle ofC×, then f is not a

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Theorem 3.1. Iff is sufficiently mild,e, g, iff satisfies (3.1), then

(3.4) d

a

dxaR[f(s)](x) =R[τaf(s)](x).

Proof. Iff is sufficiently mild, then

da

dxaR[f(s)](x) =

Z ∞ −∞

da

dxax s f(s)

Γ(1 +s)ds= Z ∞

−∞

xs−a Γ(1 +s)

Γ(1 +s−a)

f(s) Γ(1 +s)ds,

whence the variable changet=s−ayields

da

dxaR[f](x) =

Z ∞ −∞

xtf(t+a)

Γ(1 +t)dt,=R[τaf](x).

¤

Hence we have

Corollary 3.2. Under same assumption onf, we have

(3.5) log

µ d

dx

R[f(s)](x) =R

·df(s)

ds

¸ (x).

Proof. By (3.2), we infer

d da

da

dxaR[f(s)](x) =R

·

d daτaf(s)

¸ (x).

Since dadτaf(s)|a=0=dfds(s), we obtain the claimed result. ¤

Note. df(s)

ds in (3.4) is taken in the sense of distribution.. For example, iff(s) is

continuous ons≥c, differentiable ons > c, and f(s) = 0, s < c, then

log µ

d dx

R[f(s)](x) =R[f′(s)](x) + x

c

Γ(1 +c)f(c),

wheref′(s) means df(s)

ds ,s > c.

Theorem 3.1 and its Corollary show the simplest 1-parameter group (or dynamical system) {τa|a ∈R} and its generating operator d

ds are changed to the 1-parameter

group of fractional order differentiations { da

dxa|a ∈ R} and its generating operator

log¡d dx

¢

via the transformation R. Hence they suggest there may exist hierarchy of calculus involved in fractional calculus.

For the convenience, we use L[f(s)](t) =R−∞∞ estf(s)ds as the Laplace

transfor-mation in this paper. SinceLis the bilateral Laplace transformation, we have

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By definitions, we have

Similarly, we infer

(3.7) log

We also have

log

which was already shown as (2.4).

g

log

Letglog be the Lie algebra generated by log

¡d

or similar space with the Sobolev type metric as the Hilbert space on whichglogacts.

By the variable changex=etand Laplace transformation, multiplication by logx

is changed to d

ds and log

¡d

dx

¢

is changed to d ds+

space spanned by polynomials. By the variable change logx = t, Hlog is unitary

equivalent toW1/2[0,1], the Sobolev 1

2-space on [0,1]. Hence it is natural to consider

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Therefore we can saygΨ is generated by dsd and polygamma functions Ψ(m)(1 +s),

m= 0,1, . . .. Since ·

d ds,Ψ

(m)(s)

¸

= Ψ(m+1)(s), [Ψ(m)(s),Ψ(n)(s)] = 0,

denoting I(Ψm)the subspace ofgΨspanned by Ψ(m)(s),Ψ(m+1)(s), . . ., I(Ψm)is an abelian

ideal ofgΨ,m= 0,1, . . .. By definition, we have

I(Ψm)⊃I(Ψm+1),

\

m=0

I(Ψm)={0}.

We also have ·

d ds,I

(m) Ψ

¸

= I(Ψm+1).

Hence we have

dim(I(Ψm)/I(Ψm+1)) = 1.

Therefore we obtain

dim(gΨ/I(Ψm)) =m+ 1.

gΨ/I(1)Ψ is an abelian Lie algebra and the class of Ψ(1) in gΨ/I(2)Ψ is the basis of the

center. HencegΨ/I(2)Ψ is isomorphic to Heisenberg Lie algebra.

LetιΨ

log:g|log∼=gΨ be the isomorphism defined by

ιΨlog(logx) =

d ds, ι

Ψ log(log

µ

d dx

¶ ) = d

ds+ Ψ

(0)(1 +s),

and letιlogΨ = (ιΨ

log)−1. We set I (m) log =ι

log Ψ (I

(m)

Ψ . Then we obtain

Theorem 4.2. glog has a descending chain of abelian ideals I(0)log ⊃I (1)

log ⊃ · · · such that

(4.1) hlogx,I(logm)i= I(logm+1),

\

m−0

I(logm)={0}.

We have dim(glog/I(logm)) = m+ 1, m = 0,1, . . .. glog/I(logm) is abelian Lie algebras if

m= 0and 1. If m= 2, it is isomorphic to Heisenberg Lie algebra.

glog can be regarded as a kind of logarithm of Heisenberg Lie algebra. logxis a

(deformed) creation operator if we considerglog acts on Hlog. But log

¡d dx

¢

is not a (deformed) annihilation operator. To get (deformed) annihilation operator, we need to replace log¡dxd ¢bydlog= log¡dxd¢+ logx+γ. The Lie algebragdlog generated by

logxanddlog is isomorphic toglog. glog andgdlog are different. But we have

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If we do not demand generators of Heisenberg Lie algebra to be creation and an-nihilation operators, Heisenberg Lie algebra has generators such as d

dx +x, d dx −x.

Since

log µ d

dx ±x

¶ =e∓x

2 2 · d

dx·e

±x2

2 ,

the Lie algebra generated bye−x2/2

· d dx·e

x2/2

andex2/2

· d dx·e

−x2/2

is another candidate of logarithm of Heisenberg Lie algebra. It is not yet known whether this algebra is isomorphic toglog or not.

Ifξ∈gΨ, thenξis uniquely written asc0dsd +Pm≥0cnΨ(m)(1 +s), and we have

"

a0

d ds+

X

m

amΨ(m), b0

d ds+

X

m

bmΨ (m)

#

=X

m

(a0bm−amb0)Ψ(m+1).

Hence (semi) norm completions ofgΨ (andglog) become Lie algebras. For example,

ℓ2-completion g

Ψ,ℓ2 ofgΨ defined by

gΨ,ℓ2 =

(

c0

d ds+

X

m=0

cmΨ(m)

¯ ¯ ¯ ¯ ¯

|c0|2+

X

m=0

|cm|2<

)

,

is a Lie algebra having the structure of Hilbert space. Study in this direction is a future problem.

g

log

a

dxa and ealogx = xa, first we study the group Glog generated by

1-parameter groups{dxdaa|a∈R}and {xa|a∈R}.

By the variable changex=etand Laplace transformation, the operatorsxaacting

by multiplication, and da

dxa are changed to τa :τaf(s) =f(s+a) andτa

³ Γ(1+s)

Γ(1+s−a)

´ .

We setGΓ the group generated by{Γ(1+Γ(1+ss)a)|a∈R} by multiplication. a∈Racts

onGΓ bya·f =τa(f).

Proposition 1. We have

(5.1) Glog ∼=R ⋉GΓ.

SinceGΓis an abelian groupGlog is a solvable group of derived length 1. By the map

ιΣ(f(s)) =τ0f(s),

GΣis embedded isomorphically inGlog. ιΣ(GΣ) is a normal subgroup ofGlog and we

have

Glog/ιΣ(GΣ)∼=R.

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GΓ is an abelian group. But its structure seems complicated. For example, since

Γ(1 +s) Γ(1 +s−b)

µ

Γ(1 +s) Γ(1 +s−a)

¶−1

= Γ(1 +s−a) Γ(1 +s−b),

real coefficients rational function having only real roots and poles belong toGΓ. This

equality also shows definition of GΓ in this paper coincides our previous definition

ofGΓ in [2], where GΓ is defined as the group generated by {Γ(1+Γ(1+ssab))|a, b∈ R} by

multiplication.

Note. In this paper, we work in real category. If we work in complex category, then

GΓ should be the group generated by Γ(1+Γ(1+ss)a)|a∈Cby multiplication. In this case,

GΓ contains all non zero rational functions.

To study the group generated by exponential image of glog, it is convenient to

use gΨ instead of glog. We set GΨ(m) the group generated by eaΨ (m)

;a ∈ R by

multiplication and actions ofτa, a∈R. By usingGΨ(m), we define an abelian group

GΨ∞ by

GΨ∞ =

Y

m≥0

GΨ(m).

ThenGΨ∞ is an abelian group. a∈Racts as the translation operator τ

a on GΨ∞.

The groupGΨ generated by exponential image ofgΨ is written as follows:

(5.2) GΨ∼=R ⋉GΨ∞.

By (5.1), we have

Theorem 5.1. The group generated by exponential image of glog is isomorphic to

GΨ. Hence it is a solvable group of derived length 1.

Similar toGΓ, we can regardGΨ∞to be a normal subgroup ofG

Ψ. It is an abelian

group, but seems to have complicated structure. Since it is an infinite product of abelian groups, we must consider its topology. Then (together with the topology of gΨ (orglog)), it may be possible to investigategΨas the Lie algebra ofGΨ. This will

be a next problem.

Remarks on higher dimensional case

If we takex1, . . . , xn and ∂x1, . . . ,∂xn∂ as generator of Heisenberg Lie algebrahn, the

Lie algebra generated by logx1, . . . ,logxn and

log( ∂

∂x1), . . . ,log(

∂xn) is isomorphic to

n

z }| {

glog⊗ · · · ⊗glog. But if the matrix (aij) is

regular, P

ja1jxj, . . . ,Pjan,jxjandPja1,j∂xj∂ , . . . ,Pjan,j∂xj∂ are alternative generators of

hn. They are also alternative creation and annihilation operators. In this case, we

need to compute log(Pjaij∂xj∂ ). Computation of this kind of operators are done as

follows. We take ∂x∂ +∂y∂ as the example. We rewrite

∂ ∂x +

∂ ∂y =e

−x∂ ∂y

µ

∂ ∂x

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Since

Hence we have

(6.1) log

Similarly, we obtain

(6.2)

By the repeating use of this equality, we obtain

(6.3) τy;−x

∂y commute, rewriting

µ

and use Taylor expansion, we obtain another expression of (∂x∂ + ∂y∂ )a. Similar

expression of log(∂x∂ +∂y∂ ) is also possible. But these expressions seem much more complicated than (5.2) and (6.1).

Appendix. Alternative proof of Theorem 3.1

In this Appendix, we sketch an alternative proof of Theorem 3.1. In this proof, the integral transformationRappears naturally.

First we note that, since

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Since d da(x

a da

dxa)|a=0= logx+ log

¡d

dx

¢ and

d da

µ Γ(1 +X) Γ(1 +X−a)

¶¯ ¯ ¯ ¯

a=0

= Γ

(1 +X)

Γ(1 +X),

we obtain µ

logx+ log µ d

dx

¶¶

f(x) ¯ ¯ ¯ ¯

x=et

= Γ

(1 +X)

Γ(1 +X) ¯ ¯ ¯ ¯

X=d dt

f(et),

which recovers (2.4). Hence we obtain alternative proof of formulae of da and dlog

given in§3. Therefore by using Laplace transformationL[f(s)](t) =R−∞∞ estf(s)ds, we obtain (3.5) and (3.6).

Sinceτa

³ Γ(1+s)

Γ(1+s−a)f(s)

´

=Γ(1+1s)τa(Γ(1 +s)f(s)), we obtain

da

dxa

¯ ¯ ¯ ¯

x=et L

·

f(s) Γ(1 +s)

¸ (t) =L

·

τaf(s)

Γ(1 +s) ¸

(t).

Hence we have Theorem 3.1 by the variable changex=et.

References

[1] M. Abramowitz, I.A. Stegun,Handbook of Mathematical Functions, Dover Publ., New York, 1972.

[2] R. Almeida, A. Malinowska, D. Torres, A fractional calculus of variations for multiple integrals with applications to vibrating strings, arXiv:1001, 2772, Jour. Math. Phys. 51, 3 (2010), 033503-033503-12.

[3] A. Asada, Fractional calculus and Gamma function, in (Eds.: T. Iwai, S. Tan-imura, Y. Yamaguchi), ”Geometric Theory of Dynamical System and Related Topics”, Publ. RIMS 1692, Kyoto, 2010, 17-38.

[4] A. Asada,Fractional calculus and infinite order differential operator, Yokohama J. Math. 55 (2010), 129-147.

[5] J. Mikusinski,Operational Calculus, Pergamon Press, 1959.

[6] N. Nakanishi,Logarithm type functions of the differential operator, Yokohama J. Math. 55 (2009), 149-163.