Mittag-Leffler polynomials

Short description: Mathematical functions

Not to be confused with Mittag-Leffler function.

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.

Definition and examples

Generating functions

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

$\sum _{n=0}^{\mathrm{\infty }}{g}_n(x)t^n:=\frac{1}{2}(\frac{1+t}{1-t}{)}^x$ and

$\sum _{n=0}^{\mathrm{\infty }}{M}_n(x)\frac{t^n}{n!}:=(\frac{1+t}{1-t}{)}^x=(1+t{)}^x(1-t{)}^{-x}=\exp (2x\text{ artanh }t).$

They also have the bivariate generating function

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

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.

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

Properties

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

Explicit formulas

Explicit formulas are

$g_n(x)=\sum _{k=1}^n{2}^{k-1}(\genfrac{}{}{0ex}{}{n-1}{n-k})(\genfrac{}{}{0ex}{}{x}{k})=\sum _{k=0}^{n-1}{2}^k(\genfrac{}{}{0ex}{}{n-1}{k})(\genfrac{}{}{0ex}{}{x}{k+1})$

$g_n(x)=\sum _{k=0}^{n-1}(\genfrac{}{}{0ex}{}{n-1}{k})(\genfrac{}{}{0ex}{}{k+x}{n})$

$g_n(m)=\frac{1}{2}\sum _{k=0}^m(\genfrac{}{}{0ex}{}{m}{k})(\genfrac{}{}{0ex}{}{n-1+m-k}{m-1})=\frac{1}{2}\sum _{k=0}^{\min (n,m)}\frac{m}{n+m-k}(\genfrac{}{}{0ex}{}{n+m-k}{k,n-k,m-k})$

(the last one immediately shows $n{g}_n(m)=m{g}_m(n)$, a kind of reflection formula), and

$M_n(x)=(n-1)!\sum _{k=1}^{n}k{2}^k(\genfrac{}{}{0ex}{}{n}{k})(\genfrac{}{}{0ex}{}{x}{k})$, which can be also written as

$M_n(x)=\sum _{k=1}^n{2}^k(\genfrac{}{}{0ex}{}{n}{k})(n-1{)}_{n-k}(x{)}_k$, where $(x{)}_n=n!(\genfrac{}{}{0ex}{}{x}{n})=x(x-1)\cdots (x-n+1)$ 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 $m,n\in \mathbb{\mathbb{N}}$, we have the reflection formula $n{g}_n(m)=m{g}_m(n)$.

Recursion formulas

The polynomials $M_n(x)$ can be defined recursively by

$M_n(x)=2x{M}_{n-1}(x)+(n-1)(n-2)M_{n-2}(x)$, starting with $M_{-1}(x)=0$ and $M_0(x)=1$.

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

$M_{n+1}(x)=2x\sum _{k=0}^{\lfloor n/2\rfloor }\frac{n!}{(n-2k)!}{M}_{n-2k}(x)$, again starting with $M_0(x)=1$.

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

$(1)g_n(x+1)-g_{n-1}(x+1)=g_n(x)+g_{n-1}(x)$

$(2)(n+1)g_{n+1}(x)-(n-1)g_{n-1}(x)=2x{g}_n(x)$

$(3)x(g_n(x+1)-g_n(x-1))=2n{g}_n(x)$

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

Concerning recursion formula (3), the polynomial $g_n(x)$ 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 $x\in \mathbb{\mathbb{N}}$, 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 $n{g}_n(m)=m{g}_m(n)$ with respect to the main diagonal, e.g. $3\cdot 44=4\cdot 33$.

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

Orthogonality relations

For $m,n\in \mathbb{\mathbb{N}}$ the following orthogonality relation holds:

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\frac{g_n(-iy)g_m(iy)}{y\sinh \pi y}dy=\frac{1}{2n}{\delta }_{mn}.$

(Note that this is not a complex integral. As each $g_n$ 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. This has been one of the reasons to introduce the § Reduced Mittag-Leffler polynomials.)

Binomial identity

Being a Sheffer sequence of binomial type, the Mittag-Leffler polynomials $M_n(x)$ also satisfy the binomial identity

$M_n(x+y)=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})M_k(x)M_{n-k}(y)$.

Rodrigues-type formula

The classical Mittag–Leffler polynomials $g_n(x)$ satisfy a Rodrigues' formula involving the central difference operator. This formula can be derived using their connection to the Meixner–Pollaczek polynomials.

The classical Mittag–Leffler polynomials $g_n(x)$ are given by the Rodrigues-type formula

$g_n(x)=\frac{2^{n}n!}{xw(x,1)}{\delta }^n[w(x,n)]$,

where $\delta$ is the central difference operator defined by

$\delta f(z)=f(z+\frac{1}{2})-f(z-\frac{1}{2})$

and higher powers ${\delta }^n$ are obtained by successive application. The weight-like function is the Gamma ratio

$w(x,n)=\frac{\mathrm{\Gamma }(n+\frac{1}{2}-x)}{\mathrm{\Gamma }(n+\frac{1}{2}+x)}$.

This representation is obtained from the relation of $g_n(x)$ to the Meixner–Pollaczek polynomials $P_n^{(1)}(ix;\pi /2)$ and the known Rodrigues formula for the latter family.

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.

$(26)g_n(z)=\frac{\sin (\pi z)}{2\pi }\underset{-1}{\overset{1}{\int }}{t}^{n-1}(\frac{1+t}{1-t}{)}^{z}dt$

$(27)g_n(z)=\frac{\sin (\pi z)}{2\pi }\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{uz}\frac{(\tanh \frac{u}{2}{)}^n}{\sinh u}du$

$(32)g_n(z)=\frac{1}{\pi }\underset{0}{\overset{\pi }{\int }}{\cot }^z(\frac{u}{2})\cos (\frac{\pi z}{2})\cos (nu)du$

$(33)g_n(z)=\frac{1}{\pi }\underset{0}{\overset{\pi }{\int }}{\cot }^z(\frac{u}{2})\sin (\frac{\pi z}{2})\sin (nu)du$

$(34)g_n(z)=\frac{1}{2\pi }\underset{0}{\overset{2\pi }{\int }}(1+e^{it}{)}^z(2+e^{it}{)}^{n-1}{e}^{-int}dt$.

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 ${\tan }^{\pm n}$ or ${\tanh }^{\pm n}$, 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 $n⩾m⩾2$

$I(n,m):=\underset{0}{\overset{1}{\int }}\frac{{\text{artanh}}^{n}x}{x^m}dx=\underset{0}{\overset{1}{\int }}{\log }^{n/2}\left(\frac{1+x}{1-x}\right)\frac{dx}{x^m}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{z}^n\frac{{\coth }^{m-2}z}{{\sinh }^{2}z}dz.$

These integrals have the closed form

$(1)I(n,m)=\frac{n!}{2^{n-1}}{\zeta }^{n+1}{g}_{m-1}(\frac{1}{\zeta })$

in umbral notation, meaning that after expanding the polynomial in $\zeta$, each power ${\zeta }^k$ has to be replaced by the zeta value $\zeta (k)$. E.g. from $g_6(x)=\frac{1}{45}(23x^2+20x^4+2x^6)$ we get $I(n,7)=\frac{n!}{2^{n-1}}\frac{23\zeta (n-1)+20\zeta (n-3)+2\zeta (n-5)}{45}$ for $n⩾7$.

2. Likewise take for $n⩾m⩾2$

$J(n,m):=\underset{1}{\overset{\mathrm{\infty }}{\int }}\frac{{\text{arcoth}}^{n}x}{x^m}dx=\underset{1}{\overset{\mathrm{\infty }}{\int }}{\log }^{n/2}\left(\frac{x+1}{x-1}\right)\frac{dx}{x^m}=\underset{0}{\overset{\mathrm{\infty }}{\int }}{z}^n\frac{{\tanh }^{m-2}z}{{\cosh }^{2}z}dz.$

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

$(2)J(n,m)=\frac{n!}{2^{n-1}}{\eta }^{n+1}{g}_{m-1}(\frac{1}{\eta })$.

3. The following holds for $n⩾m$ with the same umbral notation for $\zeta$ and $\eta$, and completing by continuity $\eta (1):=\ln 2$.

$(3)\underset{0}{\overset{\pi /2}{\int }}\frac{x^n}{{\tan }^{m}x}dx=\cos \left(\frac{m}{2}\pi \right)\frac{(\pi /2{)}^{n+1}}{n+1}+\cos \left(\frac{m-n-1}{2}\pi \right)\frac{n!m}{2^n}{\zeta }^{n+2}{g}_m(\frac{1}{\zeta })+\sum _{v=0}^n\cos \left(\frac{m-v-1}{2}\pi \right)\frac{n!m{\pi }^{n-v}}{(n-v)!2^n}{\eta }^{n+2}{g}_m(\frac{1}{\eta }).$

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

$\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{{\arctan }^{n}x}{x^m}dx=\underset{0}{\overset{\pi /2}{\int }}\frac{x^n}{{\tan }^{m}x}dx+\underset{0}{\overset{\pi /2}{\int }}\frac{x^n}{{\tan }^{m-2}x}dx.$

4. For $n⩾m⩾2$, define $K(n,m):=\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{{\tanh }^n(x)}{x^m}dx$.

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

$(4)K(n,m):=\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{{\tanh }^n(x)}{x^m}dx=\frac{n\cdot 2^{m-1}}{(m-1)!}(-h{)}^{m-1}{g}_n(h).$

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

$K(5,3)=-\frac{2}{3}(3h_3+10h_5+2h_7)=-7\frac{\zeta (3)}{{\pi }^2}+310\frac{\zeta (5)}{{\pi }^4}-1905\frac{\zeta (7)}{{\pi }^6},$

$K(6,2)=\frac{4}{15}(23h_3+20h_5+2h_7),K(6,4)=\frac{4}{45}(23h_5+20h_7+2h_9).$

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

$(5)K(n,m)=\underset{0}{\overset{\mathrm{\infty }}{\int }}\frac{{\tanh }^n(x)}{x^m}dx=\frac{n\cdot 2^m}{(m-1)!}(-s{)}^{m-2}{g}_n(s)$, e.g.

$K(5,4)=\frac{8}{9}(3s_3+10s_5+2s_7),K(6,3)=-\frac{8}{15}(23s_3+20s_5+2s_7),K(6,5)=-\frac{8}{45}(23s_5+20s_7+2s_9).$

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

$K(5,4)=\frac{8}{9}(3s_3+10s_5+2s_7)=\frac{1}{9}\{\frac{1643}{420}-\frac{16}{315}\ln 2+3\frac{{\zeta }^{\prime }(4)}{\zeta (4)}-20\frac{{\zeta }^{\prime }(6)}{\zeta (6)}+17\frac{{\zeta }^{\prime }(8)}{\zeta (8)}\}.$

6. For $n<m$, the same integral $K(n,m)$ diverges because the integrand behaves like $x^{n-m}$ 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.

$(6)K(n-1,n)-K(n,n+1)=\underset{0}{\overset{\mathrm{\infty }}{\int }}(\frac{{\tanh }^{n-1}(x)}{x^n}-\frac{{\tanh }^n(x)}{x^{n+1}})dx=-\frac{1}{n}+\frac{(n+1)\cdot 2^n}{(n-1)!}{s}^{n-2}{g}_n(s)$.

Reduced Mittag-Leffler polynomials

The reduced Mittag-Leffler polynomials are a family of real polynomials derived from the classical Mittag-Leffler polynomials $g_n(x)$ by means of an imaginary-argument transformation. This yields real-valued polynomials that are orthogonal on the real line with respect to a hyperbolic weight function, in contrast to the classical version which uses orthogonality on the imaginary axis. The reduced form and its properties (including finite- and infinite-order differential equations) were studied in detail by Rajković et al. (2024).

Definition

The reduced Mittag-Leffler polynomials ${\varphi }_n(x)$ for $n\in {\mathbb{\mathbb{N}}}_0$ are defined as

${\varphi }_n(x)=\frac{g_{n+1}(ix)}{i^{n+1}x}$.

where $g_n(x)$ are the classical Mittag-Leffler polynomials satisfying the generating function

${\left(\frac{1+t}{1-t}\right)}^x=\sum _{n=0}^{\mathrm{\infty }}{g}_n(x)t^n(|t|<1)$,

with initial conditions $g_0(x)=1$ and $g_1(x)=2x$. They obey the three-term recurrence relation

$(n+2){\varphi }_{n+1}(x)=2x{\varphi }_n(x)-n{\varphi }_{n-1}(x)(n\in \mathbb{\mathbb{N}})$,

with initial values

${\varphi }_0(x)=2,{\varphi }_1(x)=2x$.

The polynomials satisfy the parity relation

${\varphi }_n(-x)=(-1{)}^n{\varphi }_n(x)$.

The associated monic reduced Mittag-Leffler polynomials are given by

${\hat{\varphi }}_n(x)=\frac{(n+1)!}{2^{n+1}}{\varphi }_n(x)$,

which satisfy the monic recurrence

${\hat{\varphi }}_{n+1}(x)=x{\hat{\varphi }}_n(x)-\frac{n(n+1)}{4}{\hat{\varphi }}_{n-1}(x)(n\in \mathbb{\mathbb{N}})$,

with ${\hat{\varphi }}_0(x)=1$ and ${\hat{\varphi }}_1(x)=x$.

Explicit low-degree terms

The first few monic reduced polynomials are:

• ${\hat{\varphi }}_0(x)=1$
• ${\hat{\varphi }}_1(x)=x$
• ${\hat{\varphi }}_2(x)=x^2-\frac{1}{2}$
• ${\hat{\varphi }}_3(x)=x^3-2x$
• ${\hat{\varphi }}_4(x)=x^4-5x^2+\frac{3}{2}$
• ${\hat{\varphi }}_5(x)=x^5-10x^3+\frac{23}{2}x$

Orthogonality relation

The reduced Mittag-Leffler polynomials $\{{\varphi }_n(x){\}}_{n\in {\mathbb{\mathbb{N}}}_0}$ are orthogonal on the real line with weight function $\frac{x}{\sinh (\pi x)}$:

$\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{\varphi }_n(x){\varphi }_m(x)\frac{x}{\sinh (\pi x)}dx=\frac{2}{n+1}{\delta }_{mn}(n,m\in {\mathbb{\mathbb{N}}}_0)$.

All zeros of ${\varphi }_n(x)$ (and hence of ${\hat{\varphi }}_n(x)$) are real.

Differential equations

Every monic reduced polynomial ${\hat{\varphi }}_n(x)$ satisfies a finite-order ordinary differential equation of order $n$:

$\sum _{k=1}^n({\alpha }_k+{\beta }_{k}x)\frac{{\hat{\varphi }}_n^{(k)}(x)}{k!}-n{\hat{\varphi }}_n(x)=0$,

where the coefficients are

${\alpha }_k=\cos \frac{k\pi }{2},{\beta }_k=\sin \frac{k\pi }{2}$.

For example, for $n=4$, we have ${\hat{\varphi }}_4(x)=x^4-5x^2+\frac{3}{2}$. The coefficients are:

${\alpha }_1=0,{\beta }_1=1;{\alpha }_2=-1,{\beta }_2=0;{\alpha }_3=0,{\beta }_3=-1;{\alpha }_4=1,{\beta }_4=0$.

The differential equation becomes:

$x{{\hat{\varphi }}_4}^{\prime }(x)-\frac{1}{2}{{\hat{\varphi }}_4}^{″}(x)-\frac{x}{6}{{\hat{\varphi }}_4}^{‴}(x)+\frac{1}{24}{\hat{\varphi }}_4^{(4)}(x)-4{\hat{\varphi }}_4(x)=0$.

This can be verified by computing the derivatives: ${{\hat{\varphi }}_4}^{\prime }(x)=4x^3-10x$, ${{\hat{\varphi }}_4}^{″}(x)=12x^2-10$, ${{\hat{\varphi }}_4}^{‴}(x)=24x$, ${\hat{\varphi }}_4^{(4)}(x)=24$, and substituting into the equation.

Additionally, they satisfy the infinite-order (operator) differential equation

$(\cos D+x\sin D-(n+1)I){\hat{\varphi }}_n(x)=0(D=\frac{d}{dx},I=\text{identity operator})$,

meaning ${\hat{\varphi }}_n(x)$ is an eigenfunction of the operator $\cos D+x\sin D-I$ with eigenvalue $n$.

The operators $\sin D$ and $\cos D$ act on functions via their Taylor series. For example, for ${\hat{\varphi }}_4(x)=x^4-5x^2+\frac{3}{2}$:

$\cos D{\hat{\varphi }}_4(x)={\hat{\varphi }}_4(x)-\frac{1}{2!}{{\hat{\varphi }}_4}^{″}(x)+\frac{1}{4!}{\hat{\varphi }}_4^{(4)}(x)=x^4-5x^2+\frac{3}{2}-\frac{1}{2}(12x^2-10)+\frac{1}{24}(24)=x^4-11x^2+9$,

$\sin D{\hat{\varphi }}_4(x)={{\hat{\varphi }}_4}^{\prime }(x)-\frac{1}{3!}{{\hat{\varphi }}_4}^{‴}(x)=4x^3-10x-\frac{1}{6}(24x)=4x^3-14x$,

$x\sin D{\hat{\varphi }}_4(x)=x(4x^3-14x)=4x^4-14x^2$.

Thus:

$(\cos D+x\sin D-5I){\hat{\varphi }}_4(x)=(x^4-11x^2+9)+(4x^4-14x^2)-5(x^4-5x^2+\frac{3}{2})=0$,

confirming that ${\hat{\varphi }}_4$ is an eigenfunction with eigenvalue $n=4$.

These differential properties arise because $\{{\hat{\varphi }}_n(x)\}$ form a Sheffer sequence with exponential generating function

$\hat{G}(t,x)=\frac{4\exp (2x\arctan (t/2))}{t^2+4}=\sum _{n=0}^{\mathrm{\infty }}{\hat{\varphi }}_n(x)\frac{t^n}{n!}$,

together with the theory of Sheffer sequences.

See also

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

References

• Bateman, H. (1940), "The polynomial of Mittag-Leffler", Proceedings of the National Academy of Sciences of the United States of America 26 (8): 491–496, doi:10.1073/pnas.26.8.491, ISSN 0027-8424, PMID 16588390, PMC 1078216, Bibcode: 1940PNAS...26..491B, http://authors.library.caltech.edu/8694/1/BATpnas40.pdf 
• Mittag-Leffler, G. (1891), "Sur la représentasion analytique des intégrales et des invariants d'une équation différentielle linéaire et homogène" (in French), Acta Mathematica XV: 1–32, doi:10.1007/BF02392600, ISSN 0001-5962 
• Stankovic, Miomir S.; Marinkovic, Sladjana D.; Rajkovic, Predrag M. (2010), "Deformed Mittag–Leffler Polynomials", arXiv:1007.3612 [math.NA]