Mittag-Leffler Polynomials

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
:$\displaystyle \sum_^ g_n(x)t^n :=\frac\Bigl(\frac \Bigr)^x$ and
:$\displaystyle \sum_^ M_n(x)\frac:=\Bigl(\frac \Bigr)^x=(1+t)^x(1-t)^=\exp(2x\text t).$
They also have the bivariate generating function
:$\displaystyle \sum_^\sum_^ g_n(m)x^my^n =\frac.$

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.
:

Properties

The polynomials are related by $M_n(x)=2\cdot \, g_n(x)$ and we have $g_n(1)=1$ for $n\geqslant 1$. Also $g_(\frac12)=g_(\frac12)=\frac12\frac=\frac12\cdot \frac$.

Explicit formulas

Explicit formulas are
:$g_n(x) = \sum_ ^ 2^\binom\binom xk = \sum_ ^ 2^\binom\binom x$
:$g_n(x) = \sum_^ \binomk\binomn$
:$g_n(m) = \frac12\sum_^m \binom mk\binom=\frac12\sum_^ \frac m\binom$
(the last one immediately shows $ng_n(m)=mg_m(n)$, a kind of reflection formula), and
:$M_n(x)=(n-1)!\sum_ ^k2^k\binom nk \binom xk$, which can be also written as
:$M_n(x)=\sum_ ^2^k\binom nk(n-1)_(x)_k$, where $(x)_n = n!\binom xn = x(x-1)\cdots(x-n+1)$ denotes the falling factorial.
In terms of the Gaussian hypergeometric function, we have
:$g_n(x) = x\!\cdot _2\!F_1 (1-n,1-x; 2; 2).$

Reflection formula

As stated above, for $m,n\in\mathbb N$, we have the reflection formula $ng_n(m)=mg_m(n)$.

Recursion formulas

The polynomials $M_n(x)$ can be defined recursively by
:$M_n(x)=2xM_(x)+(n-1)(n-2)M_(x)$, starting with $M_(x)=0$ and $M_(x)=1$.
Another recursion formula, which produces an odd one from the preceding even ones and vice versa, is
:$M_(x) = 2x \sum_^ \frac M_(x)$, again starting with $M_0(x) = 1$.

As for the $g_n(x)$, we have several different recursion formulas:

:$\displaystyle (1)\quad g_n(x + 1) - g_(x + 1)= g_n(x) + g_(x)$

:$\displaystyle (2)\quad (n + 1)g_(x) - (n - 1)g_(x) = 2xg_n(x)$

:$(3)\quad x\Bigl(g_n(x+1)-g_n(x-1)\Bigr) = 2ng_n(x)$

:$(4)\quad g_(m)= g_(m)+2\sum_^g_(k)=g_(1)+g_(2)+\cdots+g_(m)+g_(m-1) +\cdots+g_(1)$

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 $x\in\mathbb N$, we can apply the reflection formula to one of the identities and then swap $x$ and $n$ to obtain the other one. (As the $g_n(x)$ are polynomials, the validity extends from natural to all real values of $x$.)

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. $3\cdot44=4\cdot33$.
:

Orthogonality relations

For $m,n\in\mathbb N$ the following orthogonality relation holds:
:$\int_^\frac dy=\frac 1\delta_.$
(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 $m$ and $n$ 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
:$M_n(x+y)=\sum_^n\binom nk M_k(x)M_(y)$.

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, some of them being even valid for complex $z$, e.g.

:$(26)\qquad g_n(z) = \frac\int _^1 t^ \Bigl(\frac\Bigr)^z dt$

:$(27)\qquad g_n(z) = \frac \int_^ e^\frac du$

:$(32)\qquad g_n(z) = \frac1\pi\int _0^\pi \cot^z (\frac u2) \cos (\frac2) \cos (nu)du$

:$(33)\qquad g_n(z) = \frac1\pi\int _0^\pi \cot^z (\frac u2) \sin (\frac2) \sin (nu)du$

:$(34)\qquad g_n(z) = \frac1\int _0^ (1+e^)^z (2+e^)^ e^dt$.

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 $\tan^$ or $\tanh^$, and the degree of the Mittag-Leffler polynomial varies with $n$. 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 $n\geqslant m \geqslant 2$
:$I(n,m):= \int _0^1\dfracdx = \int _0^1\log^\Bigl(\dfrac\Bigr)\dfrac = \int _0^\infty z^n\dfrac dz.$
These integrals have the closed form
:$(1)\quad I(n,m)=\frac\zeta^~g_ (\frac1 )$
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
$g_6(x)= (23x^2+20x^4+2x^6)\$ we get
$\ I(n,7)=\frac\frac\$ for $n\geqslant 7$.

2. Likewise take for $n\geqslant m \geqslant 2$
:$J(n,m):=\int _1^\infty\dfracdx =\int _1^\infty\log^\Bigl(\dfrac\Bigr)\dfrac = \int _0^\infty z^n\dfrac dz.$

In umbral notation, where after expanding, $\eta^k$ has to be replaced by the Dirichlet eta function $\eta(k):=\left(1-2^\right)\zeta(k)$, those have the closed form
:$(2)\quad J(n,m)=\frac \eta^~g_ (\frac1 )$.

3. The following holds for $n\geqslant m$ with the same umbral notation for $\zeta$ and $\eta$, and completing by continuity $\eta(1):=\ln 2$.
:$(3)\quad \int\limits_0^ \fracdx = \cos\Bigl(\frac\pi\Bigr)\frac +\cos\Bigl(\frac\pi\Bigr) \frac\zeta^g_m(\frac1) +\sum\limits_^n \cos\Bigl(\frac\pi\Bigr)\frac \eta^g_m(\frac1).$

Note that for $n\geqslant m \geqslant 2$, this also yields a closed form for the integrals

:$\int\limits_0^ \frac dx = \int\limits_0^ \frac dx + \int\limits_0^ \frac dx.$

4. For $n\geqslant m\geqslant 2$, define $\quad K(n,m):=\int\limits_0^\infty\dfracdx$.

If $n+m$ is even and we define $h_k:= (-1)^ \frac$, we have in umbral notation, i.e. replacing $h^k$ by $h_k$,
:$(4)\quad K(n,m):=\int\limits_0^\infty\dfracdx = \dfrac(-h)^ g_n(h).$

Note that only odd zeta values (odd $k$) occur here (unless the denominators are cast as even zeta values), e.g.
:$K(5,3)=-\frac(3h_3+10h_5+2h_7)=-7\frac+ 310 \frac -1905\frac,$
:$K(6,2)=\frac(23h_3+20h_5+2h_7),\quad K(6,4)=\frac(23h_5+20h_7+2h_9).$

5. If $n+m$ is odd, the same integral is much more involved to evaluate, including the initial one $\int\limits_0^\infty\dfracdx$. Yet it turns out that the pattern subsists if we define $s_k:=\eta'(-k)=2^\zeta(-k)\ln2-(2^-1)\zeta'(-k)$, equivalently $s_k = \frac\eta(-k)+\zeta(-k)\eta(1)-\eta(-k)\eta(1)$. Then $K(n,m)$ has the following closed form in umbral notation, replacing $s^k$ by $s_k$:
:$(5)\quad K(n,m)=\int\limits_0^\infty\dfracdx=\frac(-s)^g_n(s)$, e.g.
:$K(5,4)=\frac(3s_3+10s_5+2s_7), \quad K(6,3)=-\frac(23s_3+20s_5+2s_7),\quad K(6,5)=-\frac(23s_5+20s_7+2s_9).$

Note that by virtue of the logarithmic derivative $\frac(s)+\frac(1-s)=\log\pi-\frac\frac\left(\frac\right)-\frac\frac\left(\frac\right)$ of Riemann's functional equation, taken after applying Euler's reflection formula,or see formula (14) in https://mathworld.wolfram.com/RiemannZetaFunction.html these expressions in terms of the $s_k$ can be written in terms of $\frac$, e.g.
:$K(5,4)=\frac(3s_3+10s_5+2s_7)=\frac 19\left\.$

6. For n, the same integral $K(n,m)$ diverges because the integrand behaves like $x^$ for $x\searrow 0$. But the difference of two such integrals with corresponding degree differences is well-defined and exhibits very similar patterns, e.g.
:$(6)\quad K(n-1,n)-K(n,n+1)=\int\limits_0^\infty\left(\dfrac-\dfrac\right)dx= -\frac 1n + \fracs^g_n(s)$.

See also

* Bernoulli polynomials of the second kind
* Stirling polynomials
* Poly-Bernoulli number

References

*
*
*
*{{Citation , last1=Stankovic , first1=Miomir S. , last2=Marinkovic , first2=Sladjana D. , last3=Rajkovic , first3=Predrag M. , title=Deformed Mittag–Leffler Polynomials , language=English , year=2010 , arxiv=1007.3612

Polynomials