MITTAG-LEFFLER FUNCTIONS, LP GENERALIZED N-BALLS

MITTAG-LEFFLER FUNCTIONS, LP GENERALIZED N-BALLS
AND GEOMETRIC INTERPRETATIONS
´
GORKA GUARDIOLA MUZQUIZ
Abstract. We relate the Mittag-Leffler functions with the sum of the hypervolume of generalized lp n-balls. We derive from this result some simple
properties of the Mittag-Leffler functions and a geometrical interpretation of
the fractional derivatives (work in progress).
1. Introduction
The special function:
(1.1)
Eα (z) =
∞
X
k=0
zk
, α ∈ R, z ∈ C
Γ(1 + αk)
with C being the set of complex numbers and R the set of real numbers is called
Mittag-Leffler function [6]. This function arises naturally as the eigenfunction of
the fractional derivative operator generalizing the exponential function:
dα
Eα (z α ) = Eα (z α )
dt
This function is related to the hypervolume of a generalized lp n-ball, the ball of
P
1
constant p-norm with boundary defined by kxkp = ( xpi ) p , so the ball is kxkp =
i
P
1
( xpi ) p ≤ R. This volume, as calculated by Wang and previously by Dirichlet and
(1.2)
i
others is (see the full history of this result in the paper by Wang [4]):
Vnp (R) =
(1.3)
(2Γ( p1 + 1)R)n
Γ( np + 1)
Comparing equations 1.3 and 1.1, if p =
(1.4)
Eα (2Γ(α + 1)z) =
1
α:
∞
∞
X
X
1
(2Γ(α + 1)z)n
=
Vnα (z)
Γ(αn
+
1)
n=0
n=0
A different way to state this is that, if 0 ≤ α ≤ 1, and z ∈ R, the Mittag Leffler
z
.
function is the sum of the generalized n-balls of α = p1 and radius 2Γ(α+1)
In fact the result by Wang is stronger; the volume for hyper-ellipsoids is calculated, which is related to some generalizations of the Mittag-Leffler function.
Also, the restriction of 0 ≤ α ≤ 1 can be lifted, and then the result is not a
norm. We do not treat this case here.
Date: March 19, 2015.
1
´
GORKA GUARDIOLA MUZQUIZ
2
P
1
Instead of defining the ball as kxkp = ( xpi ) p ≤ R we could use the slightly
P p i
different definition of ball kxkpp =
xi ≤ R. Looking at equation 1.2 this is,
i
in some sense, a more natural definition (we use here W to distinguish from the
volume V defined before):
Eα (2Γ(α + 1)z α ) =
(1.5)
∞
X
1
Wnα (z)
n=0
This definition has two problems, namely, the introduction of complex roots for
negative numbers (or multiple roots even for real numbers) and the problem of
defining the limit for α = 0, which would render a constant volume independent of
the radius and hence no insight.
2. Consequences and special cases
The formula 1.4 has several consequences. The first is that the result by Pollard [3] is, of course, trivial for R > 0. Pollard result proofs that in the region
1 < α ≤ ∞ (with z restricted to R) the function is completely monotonic. When
R > 0 the volume grows with the radius. This result can be made stronger for R > 0
by realizing that the area of the generalized n-balls is proportional to the derivative
of the volume (by the coarea formula). Thus, the derivative is not only positive
but also increases monotonically. When R < 0, the non-trivial result obtained by
Pollard that the function is also monotonic implies that the sum of the volumes
for odd dimensions grows faster than the sum for even dimensions (because odd
dimensions are the ones contributing the negative term). A simple special case to
illustrate this result, when α = 1, the derivative of the even part of the exponential,
is smaller than the derivative of the odd part, i.e. sinh(x) ≤ cosh(h).
The problem stated by Freden [2], i.e. the sum of all the hyperspheres of the
same radius is a special case of the Mittag-Leffler function from [6]:
(2.1)
2
E 12 (z) = ez (1 + erf (z)) = erf cx(z)
Applying this equation, and the property that Γ( 23 ) =
obtained Freden is:
(2.2)
∞
X
n=0
Vn2 (R)
√
π
2
the volume which
√π √
= erf cx( πz)
= erf cx 2
2
Another interesting special case is the exponential:
(2.3)
E1 (z) = ez
which makes
(2.4)
E1 (2Γ(2)z) = e2Γ(2)z = e2z
the sum of all the hypervolume of all the hyperdiamonds, i.e. the balls of the
standard infinite norm kxk∞ = max{xi }. The function in α = 0 is:
(2.5)
E0 (z) =
1
1−z
MITTAG-LEFFLER FUNCTIONS, LP GENERALIZED N-BALLS AND GEOMETRIC INTERPRETATIONS
3
so the sum of the hypervolume of all the hypercubes, i.e. the balls of the 1 norm
kxk1 = Σxi is:
(2.6)
E0 (2Γ(1)z) = E0 (2z) =
1
1 − 2z
The case of the exponential is specially interesting, because the even part of the
Mittag-Leffler of 21 :
2
(2.7)
even E 12 (z) = ez
which gives us the result that for even dimensions, the sum of the volumes is:
∞
X
(2.8)
2
V2n
(z) = eπz
2
n=0
which of course is a special case of the general sum of exponential properties of the
Mittag-Leffler function from [6]:
(2.9)
m−1
1
i2πr
1 X
α (z m e m ), m ∈ N
Em
m r=0
Eα (z) =
When m = 2 and α = 1:
E1 (z) =
1
1
1
1
E 1 (z 2 ) + E 12 (−z 2 )
2 2
2
1
Making the change of variables R = z 2
1
1
E 1 (R) + E 12 (−R)
2 2
2
Seeing equations 2.7 and 2.3 the questions arises about the relationship between
β
the Mittag Leffler functions and the stretched exponential functions f (x, β) = e−x .
If instead of the even dimensions we take, in general the sum of the volumes of
dimensions which are multiples of p = 1/α, we get:
E1 (R2 ) =
∞
X
p
Vpn
(z) =
n=0
=
∞
X
(2Γ( p1 + 1)R)pn
n=0
∞
X
n=0
=e
Γ( np
p + 1)
(2Γ( p1 + 1)R)pn
n!
1
(2Γ( p
+1)z)p
1
= e(2Γ(α+1)z) α =
(2.10)
∞
X
1
f 2Γ(α + 1)z, α1
1
p
Vpn
(z) = e(2Γ(α+1)z) α
n=0
There is a deep relationship between the Mittag Leffler function and the stretched
exponential function, a particular case of the Weibull distribution. First, the function relating the central n-moments of the stretched exponential function is the
´
GORKA GUARDIOLA MUZQUIZ
4
gamma function [1]. This fact is easy to proof using a change of variables on the
Laplace transform.
Z∞
(2.11)
β
tn−1 e−t dt =
1 n
Γ
β β
0
The second and more complicated relationship between the two functions was obtained by Pollard [3], also with de Laplace transform.
3. Polytopes and simplices
The extremes, α = 1 and α = 0 are, as a consequence of their construction,
the sum of all the volumes, for all dimensions, of two (dual) families of regular
polytopes (from dual norms), hypercubes for α = 0 and cross-polytopes for α = 1
(diamond, octahedron. . . ). There is only other existing family [12] of finite regular
polytopes which exists for all finite dimensions: the regular simplex family (triangle,
tetrahedron. . . ). It is self-dual. For this family, the volume of a simplex an its dual
calculated from the inradius r and outradius R respectively is:
1
(3.1)
In =
(3.2)
On =
((n + 1)n+1 nn ) 2 rn
n!
1
((n + 1)n+1 n−n ) 2 Rn
n!
To prove these identities, we take the inequalities in [13]:
1
(3.3)
1
((n + 1)n+1 nn ) 2 rn
((n + 1)n+1 n−n ) 2 Rn
≤V ≤
n!
n!
These inequalities are tight because the ratio between the outer radius and the
inner radius for a regular simplex is need a correct cite here r/R = 1/n.
On is the volume delimited by the polyhedral self-dual norm of dimension n
(simplex are convex and thus P
define a family
P∞of norms one for each dimension).
∞
If we obtain the two series n On and n In , summing for all dimensions, one
converges and other does not. It is easy to see, neglecting the 1 in the exponent
and majoring each term of the factorial with one n:
√
(3.4)
lim In ≥ lim rn n = ∞
n→∞
n→∞
This makes sense, because the hypervolume of the simplex is contained in the crust,
which is mainly in the corners (outside of the sphere) and it grows unbounded by
r. This is a well known phenomenon of curse of dimensionality.
The other series, though, converges. The ratio test gives:
(3.5)
R
On+1
=
=0
n→∞ On
n
lim
but in any case, it is easier to observe that On is bounded by an n-sphere of radius
1, and we calculated the finite sum of the volumes of the n-spheres in equation 2.1.
MITTAG-LEFFLER FUNCTIONS, LP GENERALIZED N-BALLS AND GEOMETRIC INTERPRETATIONS
5
4. Geometrical interpretation of the fractional derivative
The equation 1.2, in combination with the fact that the Mittag-Leffler function is the sum of volumes of generalized n-balls, presents us with a geometrical
interpretation for the fractional derivative.
In this context, the integer derivative, is the operator which has as eigenvector
function sum of the volumes of crosspolytopes of constant radius as eigenvalue, so
the fractional derivative is the operator which has as eigenvector function sum of the
1
volumes of generalized n-balls of radius R α . Finding the far-reaching consequences
of this geometrical interpretation is work in progress.
References
[1] Horst Rinne, The Weibull distribution: a handbook, CRC Press, 2008.
[2] Freden E., Problem 10207: Summing a Series of Volumes., Amer. Math. Monthly 100 (1948),
882.
[3] Harry Pollard, The completely monotonic character of the Mittag-Leffler function, Bulletin
of the American Mathematical Society 54 (1948), no. 12, 1115–1116.
[4] Xianfu Wang, Volumes of generalized unit balls, Mathematics Magazine (2005), 390–395.
[5] Nick Bromer, superexponentiation, Mathematics Magazine (1987), 169–174.
[6] HJ and Mathai Haubold AM and Saxena, Mittag-Leffler functions and their applications,
Journal of Applied Mathematics 2011 (2011).
[7] John W and Achar Hanneken BN, Finite Series Representation of the Inverse Mittag-Leffler
Function, Mathematical Problems in Engineering 2014 (2014).
[8] R and Seybold Hilfer HJ, Computation of the generalized Mittag-Leffler function and its
inverse in the complex plane, Integral Transforms and Special Functions 17 (2006), no. 9,
637–652.
[9] M´
ario N Berberan-Santos, Properties of the Mittag-Leffler relaxation function, Journal of
mathematical chemistry 38 (2005), no. 4, 629–635.
[10] U Westphal, Fractional powers of infinitesimal generators of semigroups, Applications of
Fractional Calculus in Physics (2000), 131–170.
[11] Michael Anthony Maroun, Generalized Quantum Theory and Mathematical Foundations of
Quantum Field Theory, UC Riverside, 2013.
[12] Harold Scott Macdonald Coxeter, Regular polytopes, Courier Corporation, 1973.
[13] L Fejes T´
oth, Regular Figures: International Series of Monographs on Pure and Applied
Mathematics, Vol. 48, Elsevier, 2014.
Rey Juan Carlos University, Madrid, Spain
E-mail address: [email protected]