hadamard scores and order slight scores:
D α y ( t ) = 1 Γ ( n − α ) ( t d d t ) n ∫ 1 t ( ln t s ) n − α + 1 y ( s ) d s s D^\alpha y(t)=\frac{1}{\Gamma(n-\alpha)}\left(t\frac{\rm{d}}{
{\rm d}t}\right)^n\int_1^t\left(\ln\frac{t}{s}\right)^{n-\alpha+1}y(s)\frac{
{\rm d}s}{s} Dαy(t)=Γ(n−α)1(tdtd)n∫1t(lnst)n−α+1y(s)sds
Among them, α ∈ [ n − 1 , n ) , n ∈ Z + \alpha\in[n-1,n),n\in\Z^+ α∈[n−1,n),n∈Z+
Hadamard scores:
I α y ( t ) = 1 Γ ( α ) ∫ 1 t ( ln t s ) α − 1 y ( s ) d s s I^\alpha y(t)=\frac{1}{\Gamma(\alpha)}\int_1^t\left(\ln\frac{t}{s}\right)^{\alpha-1}y(s)\frac{
{\rm d}s}{s} Iαy(t)=Γ(α)1∫1t(lnst)α−1y(s)sds
Among them Γ ( ⋅ ) \Gamma(·) Γ(⋅)is the Gamma function
Γ ( x ) = ∫ 0 + ∞ t x − 1 e − t d t ( x > 0 ) \Gamma(x)=\int_{0}^{+\infty} t^{x-1} e^{-t} \mathrm{~d} t(x>0) Γ(x)=∫0+∞tx−1e−t dt(x>0)