style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" | |- ! 10 | style="text-align: right;" | 10 || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" | || style="text-align: right;" | || |- |} Orthogonality relations For the following orthogonality relation holds: (Note that this is not a complex integral. As each 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 also satisfy the binomial identity . Integral representations Based on the representation as a hypergeometric function, there are several ways of representing for directly as integrals, some of them being even valid for complex , e.g. . 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 or , 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 These integrals have the closed form in umbral notation, meaning that after expanding the polynomial in , each power has to be replaced by the zeta value . E.g. from we get for . 2. Likewise take for In umbral notation, where after expanding, has to be replaced by the Dirichlet eta function , those have the closed form . 3. The following holds for with the same umbral notation for and , and completing by continuity . Note that for , this also yields a closed form for the integrals 4. For , define . If is even and we define , we have in umbral notation, i.e. replacing by , Note that only odd zeta values (odd ) occur here (unless the denominators are cast as even zeta values), e.g. 5. If is odd, the same integral is much more involved to evaluate, including the initial one . Yet it turns out that the pattern subsists if we