Mittag-Leffler polynomials

In mathematics, the Mittag-Leffler polynomials are the polynomials g_n(x) or M_n(x) studied by Mittag-Leffler (1891).

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

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

${\displaystyle \displaystyle \sum _{n=0}^{\infty }g_{n}(x)t^{n}:={\frac {1}{2}}{\Bigl (}{\frac {1+t}{1-t}}{\Bigr )}^{x}}$ and

${\displaystyle \displaystyle \sum _{n=0}^{\infty }M_{n}(x){\frac {t^{n}}{n!}}:={\Bigl (}{\frac {1+t}{1-t}}{\Bigr )}^{x}=(1+t)^{x}(1-t)^{-x}=\exp(2x{\text{ artanh }}t).}$

They also have the bivariate generating function^[1]

${\displaystyle \displaystyle \sum _{n=1}^{\infty }\sum _{m=1}^{\infty }g_{n}(m)x^{m}y^{n}={\frac {xy}{(1-x)(1-x-y-xy)}}.}$

The first few polynomials are given in the following table. The coefficients of the numerators of the ${\displaystyle g_{n}(x)}$ can be found in the OEIS,^[2] though without any references, and the coefficients of the ${\displaystyle M_{n}(x)}$ are in the OEIS^[3] as well.

n	gn(x)	Mn(x)
0	${\displaystyle {\frac {1}{2}}}$	${\displaystyle 1}$
1	${\displaystyle x}$	${\displaystyle 2x}$
2	${\displaystyle x^{2}}$	${\displaystyle 4x^{2}}$
3	${\displaystyle {\frac {1}{3}}(x+2x^{3})}$	${\displaystyle 8x^{3}+4x}$
4	${\displaystyle {\frac {1}{3}}(2x^{2}+x^{4})}$	${\displaystyle 16x^{4}+32x^{2}}$
5	${\displaystyle {\frac {1}{15}}(3x+10x^{3}+2x^{5})}$	${\displaystyle 32x^{5}+160x^{3}+48x}$
6	${\displaystyle {\frac {1}{45}}(23x^{2}+20x^{4}+2x^{6})}$	${\displaystyle 64x^{6}+640x^{4}+736x^{2}}$
7	${\displaystyle {\frac {1}{315}}(45x+196x^{3}+70x^{5}+4x^{7})}$	${\displaystyle 128x^{7}+2240x^{5}+6272x^{3}+1440x}$
8	${\displaystyle {\frac {1}{315}}(132x^{2}+154x^{4}+28x^{6}+x^{8})}$	${\displaystyle 256x^{8}+7168x^{6}+39424x^{4}+33792x^{2}}$
9	${\displaystyle {\frac {1}{2835}}(315x+1636x^{3}+798x^{5}+84x^{7}+2x^{9})}$	${\displaystyle 512x^{9}+21504x^{7}+204288x^{5}+418816x^{3}+80640x}$
10	${\displaystyle {\frac {1}{14175}}(5067x^{2}+7180x^{4}+1806x^{6}+120x^{8}+2x^{10})}$	${\displaystyle 1024x^{10}+61440x^{8}+924672x^{6}+3676160x^{4}+2594304x^{2}}$

The polynomials are related by ${\displaystyle M_{n}(x)=2\cdot {n!}\,g_{n}(x)}$ and we have ${\displaystyle g_{n}(1)=1}$ for ${\displaystyle n\geqslant 1}$. Also ${\displaystyle g_{2k}({\frac {1}{2}})=g_{2k+1}({\frac {1}{2}})={\frac {1}{2}}{\frac {(2k-1)!!}{(2k)!!}}={\frac {1}{2}}\cdot {\frac {1\cdot 3\cdots (2k-1)}{2\cdot 4\cdots (2k)}}}$.

Explicit formulas are

${\displaystyle g_{n}(x)=\sum _{k=1}^{n}2^{k-1}{\binom {n-1}{n-k}}{\binom {x}{k}}=\sum _{k=0}^{n-1}2^{k}{\binom {n-1}{k}}{\binom {x}{k+1}}}$

${\displaystyle g_{n}(x)=\sum _{k=0}^{n-1}{\binom {n-1}{k}}{\binom {k+x}{n}}}$

${\displaystyle g_{n}(m)={\frac {1}{2}}\sum _{k=0}^{m}{\binom {m}{k}}{\binom {n-1+m-k}{m-1}}={\frac {1}{2}}\sum _{k=0}^{\min(n,m)}{\frac {m}{n+m-k}}{\binom {n+m-k}{k,n-k,m-k}}}$

(the last one immediately shows ${\displaystyle ng_{n}(m)=mg_{m}(n)}$, a kind of reflection formula), and

${\displaystyle M_{n}(x)=(n-1)!\sum _{k=1}^{n}k2^{k}{\binom {n}{k}}{\binom {x}{k}}}$, which can be also written as

${\displaystyle M_{n}(x)=\sum _{k=1}^{n}2^{k}{\binom {n}{k}}(n-1)_{n-k}(x)_{k}}$, where ${\displaystyle (x)_{n}=n!{\binom {x}{n}}=x(x-1)\cdots (x-n+1)}$ denotes the falling factorial.

In terms of the Gaussian hypergeometric function, we have^[4]

${\displaystyle g_{n}(x)=x\!\cdot {}_{2}\!F_{1}(1-n,1-x;2;2).}$

As stated above, for ${\displaystyle m,n\in \mathbb {N} }$, we have the reflection formula ${\displaystyle ng_{n}(m)=mg_{m}(n)}$.

The polynomials ${\displaystyle M_{n}(x)}$ can be defined recursively by

${\displaystyle M_{n}(x)=2xM_{n-1}(x)+(n-1)(n-2)M_{n-2}(x)}$, starting with ${\displaystyle M_{-1}(x)=0}$ and ${\displaystyle M_{0}(x)=1}$.

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

${\displaystyle M_{n+1}(x)=2x\sum _{k=0}^{\lfloor n/2\rfloor }{\frac {n!}{(n-2k)!}}M_{n-2k}(x)}$, again starting with ${\displaystyle M_{0}(x)=1}$.

As for the ${\displaystyle g_{n}(x)}$, we have several different recursion formulas:

${\displaystyle \displaystyle (1)\quad g_{n}(x+1)-g_{n-1}(x+1)=g_{n}(x)+g_{n-1}(x)}$

${\displaystyle \displaystyle (2)\quad (n+1)g_{n+1}(x)-(n-1)g_{n-1}(x)=2xg_{n}(x)}$

${\displaystyle (3)\quad x{\Bigl (}g_{n}(x+1)-g_{n}(x-1){\Bigr )}=2ng_{n}(x)}$

${\displaystyle (4)\quad g_{n+1}(m)=g_{n}(m)+2\sum _{k=1}^{m-1}g_{n}(k)=g_{n}(1)+g_{n}(2)+\cdots +g_{n}(m)+g_{n}(m-1)+\cdots +g_{n}(1)}$

Concerning recursion formula (3), the polynomial ${\displaystyle g_{n}(x)}$ is the unique polynomial solution of the difference equation ${\displaystyle x(f(x+1)-f(x-1))=2nf(x)}$, normalized so that ${\displaystyle f(1)=1}$.^[5] Further note that (2) and (3) are dual to each other in the sense that for ${\displaystyle x\in \mathbb {N} }$, we can apply the reflection formula to one of the identities and then swap ${\displaystyle x}$ and ${\displaystyle n}$ to obtain the other one. (As the ${\displaystyle g_{n}(x)}$ are polynomials, the validity extends from natural to all real values of ${\displaystyle x}$.)

The table of the initial values of ${\displaystyle g_{n}(m)}$ (these values are also called the "figurate numbers for the n-dimensional cross polytopes" in the OEIS^[6]) 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. ${\displaystyle g_{5}(3)=51=33+8+10}$. It also illustrates the reflection formula ${\displaystyle ng_{n}(m)=mg_{m}(n)}$ with respect to the main diagonal, e.g. ${\displaystyle 3\cdot 44=4\cdot 33}$.

n m	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

For ${\displaystyle m,n\in \mathbb {N} }$ the following orthogonality relation holds:^[7]

${\displaystyle \int _{-\infty }^{\infty }{\frac {g_{n}(-iy)g_{m}(iy)}{y\sinh \pi y}}dy={\frac {1}{2n}}\delta _{mn}.}$

(Note that this is not a complex integral. As each ${\displaystyle g_{n}}$ is an even or an odd polynomial, the imaginary arguments just produce alternating signs for their coefficients. Moreover, if ${\displaystyle m}$ and ${\displaystyle n}$ have different parity, the integral vanishes trivially.)

Being a Sheffer sequence of binomial type, the Mittag-Leffler polynomials ${\displaystyle M_{n}(x)}$ also satisfy the binomial identity^[8]

${\displaystyle M_{n}(x+y)=\sum _{k=0}^{n}{\binom {n}{k}}M_{k}(x)M_{n-k}(y)}$.

Based on the representation as a hypergeometric function, there are several ways of representing ${\displaystyle g_{n}(z)}$ for ${\displaystyle |z|<1}$ directly as integrals,^[9] some of them being even valid for complex ${\displaystyle z}$, e.g.

${\displaystyle (26)\qquad g_{n}(z)={\frac {\sin(\pi z)}{2\pi }}\int _{-1}^{1}t^{n-1}{\Bigl (}{\frac {1+t}{1-t}}{\Bigr )}^{z}dt}$

${\displaystyle (27)\qquad g_{n}(z)={\frac {\sin(\pi z)}{2\pi }}\int _{-\infty }^{\infty }e^{uz}{\frac {(\tanh {\frac {u}{2}})^{n}}{\sinh u}}du}$

${\displaystyle (32)\qquad g_{n}(z)={\frac {1}{\pi }}\int _{0}^{\pi }\cot ^{z}({\frac {u}{2}})\cos({\frac {\pi z}{2}})\cos(nu)du}$

${\displaystyle (33)\qquad g_{n}(z)={\frac {1}{\pi }}\int _{0}^{\pi }\cot ^{z}({\frac {u}{2}})\sin({\frac {\pi z}{2}})\sin(nu)du}$

${\displaystyle (34)\qquad g_{n}(z)={\frac {1}{2\pi }}\int _{0}^{2\pi }(1+e^{it})^{z}(2+e^{it})^{n-1}e^{-int}dt}$.

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 ${\displaystyle \tan ^{\pm n}}$ or ${\displaystyle \tanh ^{\pm n}}$, and the degree of the Mittag-Leffler polynomial varies with ${\displaystyle n}$. 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,^[10] define for ${\displaystyle n\geqslant m\geqslant 2}$

${\displaystyle I(n,m):=\int _{0}^{1}{\dfrac {{\text{artanh}}^{n}x}{x^{m}}}dx=\int _{0}^{1}\log ^{n/2}{\Bigl (}{\dfrac {1+x}{1-x}}{\Bigr )}{\dfrac {dx}{x^{m}}}=\int _{0}^{\infty }z^{n}{\dfrac {\coth ^{m-2}z}{\sinh ^{2}z}}dz.}$

These integrals have the closed form

${\displaystyle (1)\quad I(n,m)={\frac {n!}{2^{n-1}}}\zeta ^{n+1}~g_{m-1}({\frac {1}{\zeta }})}$

in umbral notation, meaning that after expanding the polynomial in ${\displaystyle \zeta }$, each power ${\displaystyle \zeta ^{k}}$ has to be replaced by the zeta value ${\displaystyle \zeta (k)}$. E.g. from ${\displaystyle g_{6}(x)={\frac {1}{45}}(23x^{2}+20x^{4}+2x^{6})\ }$ we get ${\displaystyle \ I(n,7)={\frac {n!}{2^{n-1}}}{\frac {23~\zeta (n-1)+20~\zeta (n-3)+2~\zeta (n-5)}{45}}\ }$ for ${\displaystyle n\geqslant 7}$.

2. Likewise take for ${\displaystyle n\geqslant m\geqslant 2}$

${\displaystyle J(n,m):=\int _{1}^{\infty }{\dfrac {{\text{arcoth}}^{n}x}{x^{m}}}dx=\int _{1}^{\infty }\log ^{n/2}{\Bigl (}{\dfrac {x+1}{x-1}}{\Bigr )}{\dfrac {dx}{x^{m}}}=\int _{0}^{\infty }z^{n}{\dfrac {\tanh ^{m-2}z}{\cosh ^{2}z}}dz.}$

In umbral notation, where after expanding, ${\displaystyle \eta ^{k}}$ has to be replaced by the Dirichlet eta function ${\displaystyle \eta (k):=\left(1-2^{1-k}\right)\zeta (k)}$, those have the closed form

${\displaystyle (2)\quad J(n,m)={\frac {n!}{2^{n-1}}}\eta ^{n+1}~g_{m-1}({\frac {1}{\eta }})}$.

3. The following^[11] holds for ${\displaystyle n\geqslant m}$ with the same umbral notation for ${\displaystyle \zeta }$ and ${\displaystyle \eta }$, and completing by continuity ${\displaystyle \eta (1):=\ln 2}$.

${\displaystyle (3)\quad \int \limits _{0}^{\pi /2}{\frac {x^{n}}{\tan ^{m}x}}dx=\cos {\Bigl (}{\frac {m}{2}}\pi {\Bigr )}{\frac {(\pi /2)^{n+1}}{n+1}}+\cos {\Bigl (}{\frac {m-n-1}{2}}\pi {\Bigr )}{\frac {n!~m}{2^{n}}}\zeta ^{n+2}g_{m}({\frac {1}{\zeta }})+\sum \limits _{v=0}^{n}\cos {\Bigl (}{\frac {m-v-1}{2}}\pi {\Bigr )}{\frac {n!~m~\pi ^{n-v}}{(n-v)!~2^{n}}}\eta ^{n+2}g_{m}({\frac {1}{\eta }}).}$

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

${\displaystyle \int \limits _{0}^{\infty }{\frac {\arctan ^{n}x}{x^{m}}}dx=\int \limits _{0}^{\pi /2}{\frac {x^{n}}{\tan ^{m}x}}dx+\int \limits _{0}^{\pi /2}{\frac {x^{n}}{\tan ^{m-2}x}}dx.}$

4. For ${\displaystyle n\geqslant m\geqslant 2}$, define^[12] ${\displaystyle \quad K(n,m):=\int \limits _{0}^{\infty }{\dfrac {\tanh ^{n}(x)}{x^{m}}}dx}$.

If ${\displaystyle n+m}$ is even and we define ${\displaystyle h_{k}:=(-1)^{\frac {k-1}{2}}{\frac {(k-1)!(2^{k}-1)\zeta (k)}{2^{k-1}\pi ^{k-1}}}}$, we have in umbral notation, i.e. replacing ${\displaystyle h^{k}}$ by ${\displaystyle h_{k}}$,

${\displaystyle (4)\quad K(n,m):=\int \limits _{0}^{\infty }{\dfrac {\tanh ^{n}(x)}{x^{m}}}dx={\dfrac {n\cdot 2^{m-1}}{(m-1)!}}(-h)^{m-1}g_{n}(h).}$

Note that only odd zeta values (odd ${\displaystyle k}$) occur here (unless the denominators are cast as even zeta values), e.g.

${\displaystyle K(5,3)=-{\frac {2}{3}}(3h_{3}+10h_{5}+2h_{7})=-7{\frac {\zeta (3)}{\pi ^{2}}}+310{\frac {\zeta (5)}{\pi ^{4}}}-1905{\frac {\zeta (7)}{\pi ^{6}}},}$

${\displaystyle K(6,2)={\frac {4}{15}}(23h_{3}+20h_{5}+2h_{7}),\quad K(6,4)={\frac {4}{45}}(23h_{5}+20h_{7}+2h_{9}).}$

5. If ${\displaystyle n+m}$ is odd, the same integral is much more involved to evaluate, including the initial one ${\displaystyle \int \limits _{0}^{\infty }{\dfrac {\tanh ^{3}(x)}{x^{2}}}dx}$. Yet it turns out that the pattern subsists if we define^[13] ${\displaystyle s_{k}:=\eta '(-k)=2^{k+1}\zeta (-k)\ln 2-(2^{k+1}-1)\zeta '(-k)}$, equivalently ${\displaystyle s_{k}={\frac {\zeta (-k)}{\zeta '(-k)}}\eta (-k)+\zeta (-k)\eta (1)-\eta (-k)\eta (1)}$. Then ${\displaystyle K(n,m)}$ has the following closed form in umbral notation, replacing ${\displaystyle s^{k}}$ by ${\displaystyle s_{k}}$:

${\displaystyle (5)\quad K(n,m)=\int \limits _{0}^{\infty }{\dfrac {\tanh ^{n}(x)}{x^{m}}}dx={\frac {n\cdot 2^{m}}{(m-1)!}}(-s)^{m-2}g_{n}(s)}$, e.g.

${\displaystyle K(5,4)={\frac {8}{9}}(3s_{3}+10s_{5}+2s_{7}),\quad K(6,3)=-{\frac {8}{15}}(23s_{3}+20s_{5}+2s_{7}),\quad K(6,5)=-{\frac {8}{45}}(23s_{5}+20s_{7}+2s_{9}).}$

Note that by virtue of the logarithmic derivative ${\displaystyle {\frac {\zeta '}{\zeta }}(s)+{\frac {\zeta '}{\zeta }}(1-s)=\log \pi -{\frac {1}{2}}{\frac {\Gamma '}{\Gamma }}\left({\frac {s}{2}}\right)-{\frac {1}{2}}{\frac {\Gamma '}{\Gamma }}\left({\frac {1-s}{2}}\right)}$ of Riemann's functional equation, taken after applying Euler's reflection formula,^[14] these expressions in terms of the ${\displaystyle s_{k}}$ can be written in terms of ${\displaystyle {\frac {\zeta '(2j)}{\zeta (2j)}}}$, e.g.

${\displaystyle K(5,4)={\frac {8}{9}}(3s_{3}+10s_{5}+2s_{7})={\frac {1}{9}}\left\{{\frac {1643}{420}}-{\frac {16}{315}}\ln 2+3{\frac {\zeta '(4)}{\zeta (4)}}-20{\frac {\zeta '(6)}{\zeta (6)}}+17{\frac {\zeta '(8)}{\zeta (8)}}\right\}.}$

6. For ${\displaystyle n<m}$, the same integral ${\displaystyle K(n,m)}$ diverges because the integrand behaves like ${\displaystyle x^{n-m}}$ for ${\displaystyle x\searrow 0}$. But the difference of two such integrals with corresponding degree differences is well-defined and exhibits very similar patterns, e.g.

${\displaystyle (6)\quad K(n-1,n)-K(n,n+1)=\int \limits _{0}^{\infty }\left({\dfrac {\tanh ^{n-1}(x)}{x^{n}}}-{\dfrac {\tanh ^{n}(x)}{x^{n+1}}}\right)dx=-{\frac {1}{n}}+{\frac {(n+1)\cdot 2^{n}}{(n-1)!}}s^{n-2}g_{n}(s)}$.

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

• Bateman, H. (1940), "The polynomial of Mittag-Leffler" (PDF), Proceedings of the National Academy of Sciences of the United States of America, 26 (8): 491–496, Bibcode:1940PNAS...26..491B, doi:10.1073/pnas.26.8.491, ISSN 0027-8424, JSTOR 86958, MR 0002381, PMC 1078216, PMID 16588390
• Mittag-Leffler, G. (1891), "Sur la représentasion analytique des intégrales et des invariants d'une équation différentielle linéaire et homogène", Acta Mathematica (in French), XV: 1–32, doi:10.1007/BF02392600, ISSN 0001-5962, JFM 23.0327.01
• Stankovic, Miomir S.; Marinkovic, Sladjana D.; Rajkovic, Predrag M. (2010), Deformed Mittag–Leffler Polynomials, arXiv:1007.3612