integrale iterate si combinatorica

[Dragos Fratila]

Fie \( f_1,...,f_n:[0,1]\to\mathbb{R} \) functii continue.
Definim \( I_{f_1,...,f_k} \) integrala iterata a functiilor \( f_1,...,f_k \) ca fiind \( I_{f_1,...,f_k}(t) = \int_0^t I_{f_1,...,f_{k-1}}(s)f_k(s)ds \) pentru \( k\ge 2 \) si \( I_{f_1}(t) = \int_0^t f_1(s)ds \), unde \( t\in [0,1] \).

O permutare \( \sigma\in \Sigma_n=Bij\{1,2,...,n\} \) se numeste \( (p,n-p) \) -shuffle daca \( \sigma(1)<\sigma(2)<...<\sigma(p) \) si \( \sigma(p+1)<\sigma(p+2)<...<\sigma(n) \).

Demonstrati ca
\( I_{f_1,...,f_p}I_{f_{p+1},...,f_n} = \sum_{\sigma^{-1}\in(p,n-p)-\textrm{shuffles}} I_{f_{\sigma(1)},...,f_{\sigma(n)}} \).

Folosind aceasta problemuta se pot gasi relatii/formule foarte interesante intre functiile zeta multiple. De exemplu, se poate demonstra usor ca:
\( \zeta(a)\zeta(b) = \zeta(a+b)+\zeta(a,b)+\zeta(b,a) \) pentru \( a,b\ge 2 \) sau ca
\( \zeta(3)\zeta(1,2) = \zeta(4,2)+\zeta(1,5)+\zeta(3,1,2)+\zeta(1,3,2)+\zeta(1,2,3) \).