Mittag-Leffler polynomials

===Generating functions===

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

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

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

They also have the bivariate generating function{{citation | title=see the formula section of OEIS A142978| url=https://oeis.org/A142978}}

:$\backslash displaystyle\; \backslash sum\_\{n=1\}^\{\backslash infty\}\backslash sum\_\{m=1\}^\{\backslash infty\}\; g\_n(m)x^my^n\; =\backslash frac\{xy\}\{(1-x)(1-x-y-xy)\}.$

The first few polynomials are given in the following table. The coefficients of the numerators of the $g\_n(x)$ can be found in the OEIS,{{citation | title=see OEIS A064984| url= https://oeis.org/A064984}} though without any references, and the coefficients of the $M\_n(x)$ are in the OEIS{{citation | title=see OEIS A137513| url= https://oeis.org/A137513}} as well.

:

The polynomials are related by $M\_n(x)=2\backslash cdot\{n!\}\; \backslash ,\; g\_n(x)$ and we have $g\_n(1)=1$ for $n\backslash geqslant\; 1$. Also $g\_\{2k\}(\backslash frac12)=g\_\{2k+1\}(\backslash frac12)=\backslash frac12\backslash frac\{(2k-1)!!\}\{(2k)!!\}=\backslash frac12\backslash cdot\; \backslash frac\{1\backslash cdot\; 3\backslash cdots\; (2k-1)\; \}\{2\backslash cdot\; 4\backslash cdots\; (2k)\}$.

Explicit formulas are

:$g\_n(x)\; =\; \backslash sum\_\; \{k\; =\; 1\}^\{n\}\; 2^\{k-1\}\backslash binom\{n-1\}\{n-k\}\backslash binom\; xk\; =\; \backslash sum\_\; \{k\; =\; 0\}^\{n-1\}\; 2^\{k\}\backslash binom\{n-1\}\{k\}\backslash binom\; x\{k+1\}$

:$g\_n(x)\; =\; \backslash sum\_\{k\; =\; 0\}^\{n-1\}\; \backslash binom\{n-1\}k\backslash binom\{k+x\}n$

:$g\_n(m)\; =\; \backslash frac12\backslash sum\_\{k\; =\; 0\}^m\; \backslash binom\; mk\backslash binom\{n-1+m-k\}\{m-1\}=\backslash frac12\backslash sum\_\{k\; =\; 0\}^\{\backslash min(n,m)\}\; \backslash frac\; m\{n+m-k\}\backslash binom\{n+m-k\}\{k,n-k,m-k\}$

(the last one immediately shows $ng\_n(m)=mg\_m(n)$, a kind of reflection formula), and

:$M\_n(x)=(n-1)!\backslash sum\_\; \{k\; =\; 1\}^\{n\}k2^k\backslash binom\; nk\; \backslash binom\; xk$, which can be also written as

:$M\_n(x)=\backslash sum\_\; \{k\; =\; 1\}^\{n\}2^k\backslash binom\; nk(n-1)\_\{n-k\}(x)\_k$, where $(x)\_n\; =\; n!\backslash binom\; xn\; =\; x(x-1)\backslash cdots(x-n+1)$ denotes the falling factorial.

In terms of the Gaussian hypergeometric function, we have{{cite journal | author=Özmen, Nejla | author2=Nihal, Yılmaz | name-list-style=amp | title=On The Mittag-Leffler Polynomials and Deformed Mittag-Leffler Polynomials | language=English | year=2019 | url=https://dergipark.org.tr/en/download/article-file/843805}}

:$g\_n(x)\; =\; x\backslash !\backslash cdot\; \{\}\_2\backslash !F\_1\; (1-n,1-x;\; 2;\; 2).$

As stated above, for $m,n\backslash in\backslash mathbb\; N$, we have the reflection formula $ng\_n(m)=mg\_m(n)$.

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 $M\_\{0\}(x)=1$.

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

:$M\_\{n+1\}(x)\; =\; 2x\; \backslash sum\_\{k=0\}^\{\backslash lfloor\; n/2\; \backslash rfloor\}\; \backslash frac\{n!\}\{(n-2k)!\}\; M\_\{n-2k\}(x)$, again starting with $M\_0(x)\; =\; 1$.

As for the $g\_n(x)$, we have several different recursion formulas:

:$\backslash displaystyle\; (1)\backslash quad\; g\_n(x\; +\; 1)\; -\; g\_\{n-1\}(x\; +\; 1)=\; g\_n(x)\; +\; g\_\{n-1\}(x)$

:$\backslash displaystyle\; (2)\backslash quad\; (n\; +\; 1)g\_\{n+1\}(x)\; -\; (n\; -\; 1)g\_\{n-1\}(x)\; =\; 2xg\_n(x)$

:$(3)\backslash quad\; x\backslash Bigl(g\_n(x+1)-g\_n(x-1)\backslash Bigr)\; =\; 2ng\_n(x)$

:$(4)\backslash quad\; g\_\{n+1\}(m)=\; g\_\{n\}(m)+2\backslash sum\_\{k=1\}^\{m-1\}g\_\{n\}(k)=g\_\{n\}(1)+g\_\{n\}(2)+\backslash cdots+g\_\{n\}(m)+g\_\{n\}(m-1)\; +\backslash cdots+g\_\{n\}(1)$

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 $f(1)\; =\; 1$.{{citation | title=see the comment section of OEIS A142983 | url=https://oeis.org/A142983}} Further note that (2) and (3) are dual to each other in the sense that for $x\backslash in\backslash mathbb\; N$, 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{{citation | title=see OEIS A142978| url=https://oeis.org/A142978/table}}) 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. $3\backslash cdot44=4\backslash cdot33$.

:

For $m,n\backslash in\backslash mathbb\; N$ the following orthogonality relation holds:{{cite journal | author=Stankovic, Miomir S. | author2=Marinkovic, Sladjana D. | author3=Rajkovic, Predrag M. | name-list-style=amp | title=Deformed Mittag–Leffler Polynomials | language=English | year=2010 | arxiv=1007.3612 }}

:$\backslash int\_\{-\backslash infty\}^\{\backslash infty\}\backslash frac\{g\_n(-iy)g\_m(iy)\}\{y\; \backslash sinh\; \backslash pi\; y\}\; dy=\backslash frac\; 1\{2n\}\backslash delta\_\{mn\}.$

(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.)

===Binomial identity===

Being a Sheffer sequence of binomial type, the Mittag-Leffler polynomials $M\_n(x)$ also satisfy the binomial identity{{citation | title=Mathworld entry "Mittag-Leffler Polynomial" | url=https://mathworld.wolfram.com/Mittag-LefflerPolynomial.html}}

:$M\_n(x+y)=\backslash sum\_\{k=0\}^n\backslash binom\; nk\; M\_k(x)M\_\{n-k\}(y)$.

===Integral representations===

Based on the representation as a hypergeometric function, there are several ways of representing $g\_n(z)$ for directly as integrals,{{cite journal | last1=Bateman | first1=H. | title=The polynomial of Mittag-Leffler | jstor=86958 | mr=0002381 | year=1940 | journal=Proceedings of the National Academy of Sciences of the United States of America | issn=0027-8424 | volume=26 | issue=8 | pages=491–496 | doi=10.1073/pnas.26.8.491 | pmid=16588390 | pmc=1078216 | bibcode=1940PNAS...26..491B | url=http://authors.library.caltech.edu/8694/1/BATpnas40.pdf | doi-access=free }} some of them being even valid for complex $z$, e.g.

:$(26)\backslash qquad\; g\_n(z)\; =\; \backslash frac\{\backslash sin(\backslash pi\; z)\}\{2\backslash pi\}\backslash int\; \_\{-1\}^1\; t^\{n-1\}\; \backslash Bigl(\backslash frac\{1+t\}\{1-t\}\backslash Bigr)^z\; dt$

:$(27)\backslash qquad\; g\_n(z)\; =\; \backslash frac\{\backslash sin(\backslash pi\; z)\}\{2\backslash pi\}\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; e^\{uz\}\backslash frac\{(\backslash tanh\; \backslash frac\; u2)^n\}\{\backslash sinh\; u\}\; du$

:$(32)\backslash qquad\; g\_n(z)\; =\; \backslash frac1\backslash pi\backslash int\; \_0^\backslash pi\; \backslash cot^z\; (\backslash frac\; u2)\; \backslash cos\; (\backslash frac\{\backslash pi\; z\}2)\; \backslash cos\; (nu)du$

:$(33)\backslash qquad\; g\_n(z)\; =\; \backslash frac1\backslash pi\backslash int\; \_0^\backslash pi\; \backslash cot^z\; (\backslash frac\; u2)\; \backslash sin\; (\backslash frac\{\backslash pi\; z\}2)\; \backslash sin\; (nu)du$

:$(34)\backslash qquad\; g\_n(z)\; =\; \backslash frac1\{2\backslash pi\}\backslash int\; \_0^\{2\backslash pi\}\; (1+e^\{it\})^z\; (2+e^\{it\})^\{n-1\}\; e^\{-int\}dt$.

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 $\backslash tan^\{\backslash pm\; n\}$ or $\backslash tanh^\{\backslash pm\; n\}$, 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,{{citation | title=see at the end of this question on Mathoverflow | url=https://mathoverflow.net/questions/231964/how-to-prove-that-int-0-infty-frac-textarcsinhnxxmdx-is-a-rational}} define for $n\backslash geqslant\; m\; \backslash geqslant\; 2$

:$I(n,m):=\; \backslash int\; \_0^1\backslash dfrac\{\backslash text\{artanh\}^nx\}\{x^m\}dx$

= \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^2z} dz.

These integrals have the closed form

:$(1)\backslash quad\; I(n,m)=\backslash frac\{n!\}\{2^\{n-1\}\}\backslash zeta^\{n+1\}~g\_\; \{m-1\}(\backslash frac1\{\backslash zeta\}\; )$

in umbral notation, meaning that after expanding the polynomial in $\backslash zeta$, each power $\backslash zeta^k$ has to be replaced by the zeta value $\backslash zeta(k)$. E.g. from

$g\_6(x)=\{\backslash frac\{1\}\{45\}\}\; (23x^2+20x^4+2x^6)\backslash$ we get

$\backslash \; I(n,7)=\backslash frac\{n!\}\{2^\{n-1\}\}\backslash frac\{23~\backslash zeta(n-1)+20~\backslash zeta(n-3)+2~\backslash zeta(n-5)\}\{45\}\backslash$ for $n\backslash geqslant\; 7$.

2. Likewise take for $n\backslash geqslant\; m\; \backslash geqslant\; 2$

:$J(n,m):=\backslash int\; \_1^\backslash infty\backslash dfrac\{\backslash text\{arcoth\}^nx\}\{x^m\}dx\; =\backslash int\; \_1^\backslash infty\backslash log^\{n/2\}\backslash Bigl(\backslash dfrac\{x+1\}\{x-1\}\backslash Bigr)\backslash dfrac\{dx\}\{x^m\}$

= \int _0^\infty z^n\dfrac{\tanh^{m-2}z }{\cosh^2z} dz.

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

:$(2)\backslash quad\; J(n,m)=\backslash frac\{n!\}\{2^\{n-1\}\}\; \backslash eta^\{n+1\}~g\_\; \{m-1\}(\backslash frac1\{\backslash eta\}\; )$.

3. The following{{citation | title=answer on math.stackexchange | url=https://math.stackexchange.com/a/2939452}} holds for $n\backslash geqslant\; m$ with the same umbral notation for $\backslash zeta$ and $\backslash eta$, and completing by continuity $\backslash eta(1):=\backslash ln\; 2$.

:$(3)\backslash quad\; \backslash int\backslash limits\_0^\{\backslash pi/2\}\; \backslash frac\{x^n\}\{\backslash tan^m\; x\}dx\; =\; \backslash cos\backslash Bigl(\backslash frac\{\; m\}\{2\}\backslash pi\backslash Bigr)\backslash frac\{(\backslash pi/2)^\{n+1\}\}\{n+1\}$

+\cos\Bigl(\frac{ m-n-1}{2}\pi\Bigr) \frac{n!~m}{2^{n}}\zeta^{n+2}g_m(\frac1{\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(\frac1{\eta}).

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

:$\backslash int\backslash limits\_0^\{\backslash infty\}\; \backslash frac\{\backslash arctan^n\; x\}\{x^m\}\; dx\; =\; \backslash int\backslash limits\_0^\{\backslash pi/2\}\; \backslash frac\{x^n\}\{\backslash tan^m\; x\}\; dx\; +\; \backslash int\backslash limits\_0^\{\backslash pi/2\}\; \backslash frac\{x^n\}\{\backslash tan^\{m-2\}\; x\}\; dx.$

4. For $n\backslash geqslant\; m\backslash geqslant\; 2$, define{{citation | title=similar to this question on Mathoverflow | url=https://math.stackexchange.com/q/1582943 }} $\backslash quad\; K(n,m):=\backslash int\backslash limits\_0^\backslash infty\backslash dfrac\{\backslash tanh^n(x)\}\{x^m\}dx$.

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

:$(4)\backslash quad\; K(n,m):=\backslash int\backslash limits\_0^\backslash infty\backslash dfrac\{\backslash tanh^n(x)\}\{x^m\}dx\; =$

\dfrac{n\cdot 2^{m-1}}{ (m-1)!}(-h)^{m-1} g_n(h).

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

:$K(5,3)=-\backslash frac\{2\}\{3\}(3h\_3+10h\_5+2h\_7)=-7\backslash frac\{\backslash zeta(3)\}\{\backslash pi^2\}+\; 310\; \backslash frac\{\backslash zeta(5)\}\{\backslash pi^4\}\; -1905\backslash frac\{\backslash zeta(7)\}\{\backslash pi^6\},$

:$K(6,2)=\backslash frac\{4\}\{15\}(23h\_3+20h\_5+2h\_7),\backslash quad\; K(6,4)=\backslash frac\{4\}\{45\}(23h\_5+20h\_7+2h\_9).$

5. If $n+m$ is odd, the same integral is much more involved to evaluate, including the initial one $\backslash int\backslash limits\_0^\backslash infty\backslash dfrac\{\backslash tanh^3(x)\}\{x^2\}dx$. Yet it turns out that the pattern subsists if we define{{citation | title=method used in this answer on Mathoverflow | url=https://mathoverflow.net/a/271569}} $s\_k:=\backslash eta\text{'}(-k)=2^\{k+1\}\backslash zeta(-k)\backslash ln2-(2^\{k+1\}-1)\backslash zeta\text{'}(-k)$, equivalently $s\_k\; =\; \backslash frac\{\backslash zeta(-k)\}\{\backslash zeta\text{'}(-k)\}\backslash eta(-k)+\backslash zeta(-k)\backslash eta(1)-\backslash eta(-k)\backslash eta(1)$. Then $K(n,m)$ has the following closed form in umbral notation, replacing $s^k$ by $s\_k$:

:$(5)\backslash quad\; K(n,m)=\backslash int\backslash limits\_0^\backslash infty\backslash dfrac\{\backslash tanh^n(x)\}\{x^m\}dx=\backslash frac\{n\backslash cdot2^\{m\}\}\{(m-1)!\}(-s)^\{m-2\}g\_n(s)$, e.g.

:$K(5,4)=\backslash frac\{8\}\{9\}(3s\_3+10s\_5+2s\_7),\; \backslash quad\; K(6,3)=-\backslash frac\{8\}\{15\}(23s\_3+20s\_5+2s\_7),\backslash quad\; K(6,5)=-\backslash frac\{8\}\{45\}(23s\_5+20s\_7+2s\_9).$

Note that by virtue of the logarithmic derivative $\backslash frac\{\backslash zeta\text{'}\}\{\backslash zeta\}(s)+\backslash frac\{\backslash zeta\text{'}\}\{\backslash zeta\}(1-s)=\backslash log\backslash pi-\backslash frac\{1\}\{2\}\backslash frac\{\backslash Gamma\text{'}\}\{\backslash Gamma\}\backslash left(\backslash frac\{s\}\{2\}\backslash right)-\backslash frac\{1\}\{2\}\backslash frac\{\backslash Gamma\text{'}\}\{\backslash Gamma\}\backslash left(\backslash frac\{1-s\}\{2\}\backslash right)$ 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 $\backslash frac\{\backslash zeta\text{'}(2j)\; \}\{\backslash zeta(2j)\; \}$, e.g.

:$K(5,4)=\backslash frac\{8\}\{9\}(3s\_3+10s\_5+2s\_7)=\backslash frac\; 19\backslash left\backslash \{\; \backslash frac\{1643\}\{420\}-\backslash frac\{16\; \}\{315\}\backslash ln2+3\backslash frac\{\backslash zeta\text{'}(4)\; \}\{\backslash zeta(4)\; \}-20\backslash frac\{\backslash zeta\text{'}(6)\; \}\{\backslash zeta(6)\; \}+17\backslash frac\{\backslash zeta\text{'}(8)\; \}\{\backslash zeta(8)\; \}\backslash right\backslash \}.$

6. For

:$(6)\backslash quad\; K(n-1,n)-K(n,n+1)=\backslash int\backslash limits\_0^\backslash infty\backslash left(\backslash dfrac\{\backslash tanh^\{n-1\}(x)\}\{x^\{n\}\}-\backslash dfrac\{\backslash tanh^\{n\}(x)\}\{x^\{n+1\}\}\backslash right)dx=\; -\backslash frac\; 1n\; +\; \backslash frac\{\; (n+1)\backslash cdot2^\{n\}\}\{(n-1)!\}s^\{n-2\}g\_n(s)$.

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

{{Reflist}}

• {{Citation | last1=Bateman | first1=H. | title=The polynomial of Mittag-Leffler | jstor=86958 | mr=0002381 | year=1940 | journal=Proceedings of the National Academy of Sciences of the United States of America | issn=0027-8424 | volume=26 | issue=8 | pages=491–496 | doi=10.1073/pnas.26.8.491| pmid=16588390 | pmc=1078216 | bibcode=1940PNAS...26..491B | url=http://authors.library.caltech.edu/8694/1/BATpnas40.pdf | doi-access=free }}
• {{Citation | last1=Mittag-Leffler | first1=G. | title=Sur la représentasion analytique des intégrales et des invariants d'une équation différentielle linéaire et homogène | language=French | doi=10.1007/BF02392600 | jfm=23.0327.01 | year=1891 | journal=Acta Mathematica | issn=0001-5962 | volume=XV | pages=1–32| doi-access=free }}
• {{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 }}

Category:Polynomials