Mittag-Leffler polynomials explained

Mittag-Leffler polynomials should not be confused with Mittag-Leffler function.

In mathematics, the Mittag-Leffler polynomials are the polynomials g_n(x) or M_n(x) studied by .

M_n(x) is a special case of the Meixner polynomial M_n(x;b,c) at b = 0, c = -1.

Definition and examples

Generating functions

The Mittag-Leffler polynomials are defined respectively by the generating functions

\displaystyle

	infty
\sum
	n=0

	n
g		:=
	n(x)t

	1	l(
	2

	1+t
	1-t

r)^x

and

\displaystyle

	infty
\sum
	n=0

:=l(

n(x)	tⁿ
	n!

	1+t
	1-t

r)^x=(1+t)^x(1-t)^-x=\exp(2xartanht).

They also have the bivariate generating function

\displaystyle

	infty
\sum
	n=1

	infty
\sum
	m=1

	my
g
	n(m)x

ⁿ=

	xy
	(1-x)(1-x-y-xy)

Examples

The first few polynomials are given in the following table. The coefficients of the numerators of the

g_n(x)

can be found in the OEIS, though without any references, and the coefficients of the

M_n(x)

are in the OEIS as well.

g_n(x)

M_n(x)

	1
	2

x²

4x²

{	1
	3

} (x+2x^3)

8x^3+4x

{	1
	3

} (2x^2+x^4)

16x^4+32x²

{	1
	15

} (3x+10x^3+2x^5)

32x⁵+160x³+48x

{	1
	45

} (23x^2+20x^4+2x^6)

64x⁶+640x⁴+736x²

{	1
	315

} (45 x + 196 x^3 + 70 x^5 + 4 x^7)

128x⁷+2240x⁵+6272x³+1440x

{	1
	315

} (132 x^2 + 154 x^4 + 28 x^6 + x^8)

256x⁸+7168x⁶+39424x⁴+33792x²

{	1
	2835

} (315 x + 1636 x^3 + 798 x^5 + 84 x^7 + 2 x^9)

512x⁹+21504x⁷+204288x⁵+418816x³+80640x

{	1
	14175

} (5067 x^2 + 7180 x^4 + 1806 x^6 + 120 x^8 + 2 x^)

1024x¹⁰+61440x⁸+924672x⁶+3676160x⁴+2594304x²

Properties

The polynomials are related by

M_{n(x)=2 ⋅ {n!}}g_n(x)

and we have

g_n(1)=1

for

n\geqslant1

. Also

g_2k(

	12)=g	(
	_2k+1

12)=

12	(2k-1)!!
	(2k)!!

12 ⋅

	1 ⋅ 3 … (2k-1)
	2 ⋅ 4 … (2k)

Explicit formulas

Explicit formulas are

g_n(x)=

	n
\sum
	k=1

2^k-1\binom{n-1}{n-k}\binomxk=

	n-1
\sum
	k=0

2^k\binom{n-1}{k}\binomx{k+1}

g_n(x)=

	n-1
\sum
	k=0

\binom{n-1}k\binom{k+x}n

g_n(m)=

	12\sum
	_k

^m\binommk\binom{n-1+m-k}{m-1}=

	12\sum
	_k

^min(n,m)

	m{n+m-k}\binom{n+m-k}{k,n-k,m-k}

(the last one immediately shows

ng_n(m)=mg_m(n)

, a kind of reflection formula), and

M_{n(x)=(n-1)!\sum}

	n

	k=1

k2^k\binomnk\binomxk

, which can be also written as

M_n(x)=\sum

	n

	k=1

2^k\binomnk(n-1)_n-k(x)_k

, where

(x)_n=n!\binomxn=x(x-1) … (x-n+1)

denotes the falling factorial.In terms of the Gaussian hypergeometric function, we have^[1]

g_n(x)=x ⋅ {}_2F₁(1-n,1-x;2;2).

Reflection formula

As stated above, for

m,n\inN

, we have the reflection formula

ng_n(m)=mg_m(n)

Recursion formulas

The polynomials

M_n(x)

can be defined recursively by

M_n(x)=2xM_n-1(x)+(n-1)(n-2)M_n-2(x)

, starting with

M_-1(x)=0

and

M₀(x)=1

.Another recursion formula, which produces an odd one from the preceding even ones and vice versa, is

M_n+1(x)=2x

	\lfloorn/2\rfloor
\sum
	k=0

	n!
	(n-2k)!

M_n-2k(x)

, again starting with

M_0(x)=1

. As for the

g_n(x)

, we have several different recursion formulas:

\displaystyle(1) g_n(x+1)-g_n-1(x+1)=g_n(x)+g_n-1(x)

\displaystyle(2) (n+1)g_n+1(x)-(n-1)g_n-1(x)=2xg_n(x)

(3) xl(g_n(x+1)-g_n(x-1)r)=2ng_n(x)

(4) g_n+1(m)=g_n

	m-1
(m)+2\sum
	k=1

g_n(k)=g_n(1)+g_n(2)+ … +g_n(m)+g_n(m-1)+ … +g_n(1)

Concerning recursion formula (3), the polynomial

g_n(x)

is the unique polynomial solution of the difference equation

x(f(x+1)-f(x-1))=2nf(x)

, normalized so that

f(1)=1

. Further note that (2) and (3) are dual to each other in the sense that for

x\inN

, we can apply the reflection formula to one of the identities and then swap

and

to obtain the other one. (As the

g_n(x)

are polynomials, the validity extends from natural to all real values of

Initial values

The table of the initial values of

g_n(m)

(these values are also called the "figurate numbers for the n-dimensional cross polytopes" in the OEIS) may illustrate the recursion formula (1), which can be taken to mean that each entry is the sum of the three neighboring entries: to its left, above and above left, e.g.

g_{5(3)=51=33+8+10}

. It also illustrates the reflection formula

ng_n(m)=mg_m(n)

with respect to the main diagonal, e.g.

3 ⋅ 44=4 ⋅ 33

	1	2	3	4	5	6	7	8	9	10
1	1	1	1	1	1	1	1	1	1	1
2	2	4	6	8	10	12	14	16	18
3	3	9	19	33	51	73	99	129
4	4	16	44	96	180	304	476
5	5	25	85	225	501	985
6	6	36	146	456	1182
7	7	49	231	833
8	8	64	344
9	9	81
10	10

Orthogonality relations

For

m,n\inN

the following orthogonality relation holds:^[2]

	infty
\int
	-infty

	g_n(-iy)g_m(iy)	dy=
	y\sinh\piy

	1{2n}\delta
	_mn

(Note that this is not a complex integral. As each

g_n

is an even or an odd polynomial, the imaginary arguments just produce alternating signs for their coefficients. Moreover, if

and

have different parity, the integral vanishes trivially.)

Binomial identity

Being a Sheffer sequence of binomial type, the Mittag-Leffler polynomials

M_n(x)

also satisfy the binomial identity

M_n(x+y)=\sum

	n\binom

	k=0

nkM_k(x)M_n-k(y)

Integral representations

Based on the representation as a hypergeometric function, there are several ways of representing

g_n(z)

for

|z|<1

directly as integrals,^[3] some of them being even valid for complex

, e.g.

(26) g_n(z)=

	\sin(\piz)
	2\pi

\int

	1

	-1

t^n-1l(

	1+t
	1-t

r)^zdt

(27) g_n(z)=

	\sin(\piz)
	2\pi

	infty
\int
	-infty

e^uz

(\tanh

	u2)
	ⁿ

\sinhu

(32) g_n(z)=

	1\pi\int
	₀

^\pi\cot^z(

	u2)	(
	\cos

	\piz
	2)

\cos(nu)du

(33) g_n(z)=

	1\pi\int
	₀

^\pi\cot^z(

	u2)	(
	\sin

	\piz
	2)

\sin(nu)du

(34) g_n(z)=

	1{2\pi}\int
	₀

^2\pi(1+e^it)^z(2+e^it)^n-1e^-intdt

Closed forms of integral families

There are several families of integrals with closed-form expressions in terms of zeta values where the coefficients of the Mittag-Leffler polynomials occur as coefficients. All those integrals can be written in a form containing either a factor

\tan^\pm

\tanh^\pm

, and the degree of the Mittag-Leffler polynomial varies with

. One way to work out those integrals is to obtain for them the corresponding recursion formulas as for the Mittag-Leffler polynomials using integration by parts.

1. For instance, define for

n\geqslantm\geqslant2

I(n,m):=\int

	1\dfrac{artanh

	0

^nx}{x^m}dx
=\int

	1log

	0

^n/2l(\dfrac{1+x}{1-x}r)\dfrac{dx}{x^m}
=\int

	infty

	0

z^n\dfrac{\coth^m-2z}{\sinh^2z}dz.

These integrals have the closed form

(1) I(n,m)=

	n!
	2^n-1

\zetaⁿ⁺¹~g_m-1(

	1{\zeta}
	)

in umbral notation, meaning that after expanding the polynomial in

\zeta

, each power

\zeta^k

has to be replaced by the zeta value

\zeta(k)

. E.g. from

6(x)={	1
	45

} (23x^2+20x^4+2x^6)\ we get

I(n,7)=	n!
	2^n-1

	23~\zeta(n-1)+20~\zeta(n-3)+2~\zeta(n-5)
	45

for

n\geqslant7

2. Likewise take for

n\geqslantm\geqslant2

J(n,m):=\int

	infty\dfrac{arcoth

	1

^nx}{x^m}dx=\int

	inftylog

	1

^n/2l(\dfrac{x+1}{x-1}r)\dfrac{dx}{x^m}
=\int

	infty

	0

z^{n\dfrac{\tanh}^m-2z}{\cosh^2z}dz.

In umbral notation, where after expanding,

η^k

has to be replaced by the Dirichlet eta function

η(k):=\left(1-2^1-k\right)\zeta(k)

, those have the closed form

(2) J(n,m)=

	n!
	2^n-1

ηⁿ⁺¹~g_m-1(

	1{η}
	)

3. The following holds for

n\geqslantm

with the same umbral notation for

\zeta

and

, and completing by continuity

η(1):=ln2

(3)

	\pi/2
\int\limits
	0

	xⁿ
	\tan^mx

dx=\cosl(

	m	\pir)
	2

	(\pi/2)ⁿ⁺¹	+\cosl(
	n+1

	m-n-1
	2

\pir)

	n!~m
	2ⁿ

\zetaⁿ⁺²

\cosl(

m(	1{\zeta}) +\sum\limits
	_v=0

	m-v-1	\pir)
	2

	n!~m~\pi^n-v
	(n-v)!~2ⁿ

ηⁿ⁺²

m(	1{η}).

Note that for

n\geqslantm\geqslant2

, this also yields a closed form for the integrals

	infty
\int\limits
	0

	\arctanⁿx
	x^m

dx=

	\pi/2
\int\limits
	0

	xⁿ
	\tan^mx

dx+

	\pi/2
\int\limits
	0

	xⁿ
	\tan^m-2x

dx.

4. For

n\geqslantm\geqslant2

, define

	infty\dfrac{\tanh
K(n,m):=\int\limits
	0

^n(x)}{x^m}dx

n+m

is even and we define

h_k:=

	k-1
	2

(-1)

	(k-1)!(2^k-1)\zeta(k)
	2^k-1\pi^k-1

, we have in umbral notation, i.e. replacing

h^k

h_k

(4)

	infty\dfrac{\tanh
K(n,m):=\int\limits
	0

^n(x)}{x^m}dx= \dfrac{n ⋅ 2^m-1

}(-h)^ g_n(h). Note that only odd zeta values (odd

) occur here (unless the denominators are cast as even zeta values), e.g.

K(5,3)=-	2
	3

(3h_3+10h_5+2h

7)=-7	\zeta(3)
	\pi²

+310

	\zeta(5)	-1905
	\pi⁴

	\zeta(7)
	\pi⁶

K(6,2)=	4
	15

(23h_3+20h_5+2h_7),K(6,4)=

	4
	45

(23h_5+20h_7+2h_9).

5. If

n+m

is odd, the same integral is much more involved to evaluate, including the initial one

	infty\dfrac{\tanh
\int\limits
	0

^3(x)}{x^2}dx

. Yet it turns out that the pattern subsists if we define

	k+1
s
	k:=η'(-k)=2

\zeta(-k)ln2-(2^k+1-1)\zeta'(-k)

, equivalently

s_k=

	\zeta(-k)
	\zeta'(-k)

η(-k)+\zeta(-k)η(1)-η(-k)η(1)

. Then

K(n,m)

has the following closed form in umbral notation, replacing

s^k

s_k

(5)

	infty\dfrac{\tanh
K(n,m)=\int\limits
	0

^n(x)}{x

m}dx=	n ⋅ 2^m
	(m-1)!

(-s)^m-2g_n(s)

, e.g.

K(5,4)=	8
	9

(3s_3+10s_5+2s_7), K(6,3)=-

	8
	15

(23s_3+20s_5+2s_7),K(6,5)=-

	8
	45

(23s_5+20s_7+2s_9).

	\zeta'	(s)+
	\zeta

	\zeta'	(1-s)=log\pi-
	\zeta

	1
	2

	\Gamma'	\left(
	\Gamma

	s	\right)-
	2

	1
	2

	\Gamma'	\left(
	\Gamma

	1-s
	2

\right)

of Riemann's functional equation, taken after applying Euler's reflection formula,^[4] these expressions in terms of the

s_k

can be written in terms of

	\zeta'(2j)
	\zeta(2j)

, e.g.

K(5,4)=	8
	9

(3s_3+10s_5+2s

7)=

19\left\{

	1643
	420

	16	ln2+3
	315

	\zeta'(4)	-20
	\zeta(4)

	\zeta'(6)	+17
	\zeta(6)

	\zeta'(8)
	\zeta(8)

\right\}.

6. For

n<m

, the same integral

K(n,m)

diverges because the integrand behaves like

x^n-m

for

x\searrow0

. But the difference of two such integrals with corresponding degree differences is well-defined and exhibits very similar patterns, e.g.

(6)

	infty\left(\dfrac{\tanh
K(n-1,n)-K(n,n+1)=\int\limits
	0

^n-1(x)}{xⁿ

}-\dfrac\right)dx= -\frac 1n + \fracs^g_n(s) .

Notes and References

Özmen, Nejla . Nihal, Yılmaz . amp . On The Mittag-Leffler Polynomials and Deformed Mittag-Leffler Polynomials . English . 2019 .
Stankovic, Miomir S. . Marinkovic, Sladjana D. . Rajkovic, Predrag M. . amp . Deformed Mittag–Leffler Polynomials . English . 2010 . 1007.3612 .
Bateman . H. . The polynomial of Mittag-Leffler . 86958 . 0002381 . 1940 . . 0027-8424 . 26 . 8 . 491–496 . 10.1073/pnas.26.8.491 . 16588390 . 1078216 . 1940PNAS...26..491B . free .
or see formula (14) in https://mathworld.wolfram.com/RiemannZetaFunction.html

Mittag-Leffler polynomials explained

Definition and examples

Generating functions

Examples

Properties

Explicit formulas

Reflection formula

Recursion formulas

Initial values

Orthogonality relations

Binomial identity

Integral representations

Closed forms of integral families

See also

Notes and References