FOR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER

34 T. S. Aleroev, H. T. Aleroeva, Ning-Ming Nie, and Yi-Fa Tangfunctions D a (A) and E ρ (λ; ρ −1 ) are of genus zero, we will notice thatD A (λ) = cE ρ (λ; 2),where c is a constant unknown as yet. Consider the logarithmic derivative[ln DA (λ ] ′ D ′=A (λ)∞D A (λ) = − ∑ λ j∞∑ ∞∑= (λ k j k)λ k−1 (|λ| < λ −111 − λλ ),jj=1k=1 j=1that is,[ln DA (λ) ] ′=D ′ A (λ)D A (λ) = −sp( A(J − λA) −1) = − ∑ χ n+1λ n ,where χ n= spA n .Let D A (A) = ∞ ∑0=1a k λ k be the representation by Taylor’s series of thefunction D A (A). Establish the interrelation between χ kand a k . AsD A (λ) = E ρ (λ; ρ −1 ) = ∑ λ kΓ(ρ −1 + kρ −1 ) ,1we have a n =Γ(ρ −1 +kρ −1 ). Further, sincewe obtain the recurrent formulaD ′ A (λ)D A (λ) = ∑ (nan λ n )∑an λ n= − ∑ χ n+1λ n ,χ n+1+ ∑ a s χ n+1−s= −(n + 1)a n+1 ,χ n+1=− ∑ [a n−k χ k−na n = − ∑ (−1) n−k]Γ(2−β(n − k)) χ −n (−1)n ,kΓ(2−nβ)1χ 1=Γ(2 + 1/ρ) , 2a 2 + a 1 χ 1= −χ 2, χ 2= a 2 1 − 2a 2 ,2∑−χ 3= 3a 2 3χ 3−s, χ 3= a 2 a 1 + a 1 (2a 2 − a 2 1) − 3a 2 .s=1As 1/χ 1< λ 1 < χ 1/χ 2, we have1Γ(2 + 1/ρ) < λ 21 0.j=0