Symptotic expansions for alternating CFS. By way of example, using the Equation (163), the
Symptotic expansions for alternating CFS. As an example, employing the Equation (163), the following expression can be derived:n -1 Fr k =(-1)k log1+k nN0 t N t log 1 + + (-1)n+1 0 log 1 + 0! n t =0 0! n n+1 log(two) = (-1) .t=n(164)To cover the general case, Alabdulmohsin [16] proposed an expansion of Formula (163) -1 to FFS together with the kind F r n=0 ei g(), where the exponentials are complicated, i.e., with 0 2, by f G (n) =r ( r ) r in r g (0) – e g (n) , r! r! r =0 r =where r ==ei r .(165)Mathematics 2021, 9,31 ofAdditionally, it holds that f G (n) ==ei g() – F r ei g() .=n(166)-1 Thinking of an OSFS from the form F r n=0 s g(), exactly where (s )N = (s0 , s1 , ) is a periodic sequence of indicators, with period p, then 0 s = 0 implies that (indeed: if and = only if) the series of kind 0 s r are T-summable. The T-sum Sr = 0 s r might be = = obtained in the recurrence equationS0 =1 pp -1 =skk =,andFrr -1 =r (r -) p S + s r = 0 , =p -(167)and, for all n N, the function f (n) may be written as f (n) =1 1 Sr (0) g(r) (0) – Sr n mod ( p) g(r) (n) . r! r! r =0 r =(168)Moreover, when the function g(n) features a finite polynomial order m, then it holds that=pn-s g() = limn=s g() +Sr ( 0 ) ( r ) g ( pn) . r! r =m(169)As a consequence of Equation (168), the discrete Fourier transform [140,141] can be made use of to CFT8634 Technical Information locate the exclusive natural expansion f G (n) for an OSFS on the type f (n) = F r n-1 s g , exactly where s can be a periodic sign sequence, yielding: =0 f (n) = w0 F r g() + w1 F r ei/p g() + + w p-1 F r ei( p-1)/p g() ,=0 =0 =0 p -1 n -1 n -1 n -(170)where the weights wm are defined by wm = 1 =0 s e-im/p . Furthermore, the special p generalization f G (n) of f (n), that is consistent together with the polynomial approximation method presented in (140), is the sum with the one of a kind organic expansions for every term in Equation (170). 4.3.6. Procedures to Evaluate Fractional Finite Sums According to Alabdulmohsin, the EMSF and its analogous can not be utilised to calculate the FFS directly, because they can diverge quickly. Moreover, computing the Taylor series expansions and utilizing them to calculate an FFS are laborious tasks. The sensible strategies introduced by Alabdulmohsin to obtain the values of FFS using the kind -1 f (n) = F r n=0 s g(, n) for all n C are presented inside the following section. -1 For SFS in the variety f (n) = F r n=0 g(), exactly where g is analytic having a finite polynomial order m, the system discussed in Section 4.two may be applied. Alabdulmohsin reports that the M ler chleicher BMS-8 supplier technique is actually a specific case for evaluating arbitrary SFS.-1 For an SFS on the form f (n) = F r n=0 g(), if f G (n) could be the special all-natural expansion in line with Equation (140), then, for all n C, the function f G (n) is often written ass -1 br (n) (r) g (s) + F r g() – g( + n) , r! r =0 =f G (n) =(171)Mathematics 2021, 9,32 of-1 exactly where Equation (171) holds for all s C and br (n) = F r n=0 r is really a polynomial offered by the Faulhaber’s formula (59). In addition, in the event the function g includes a finite polynomial order m, then f G (n) may be evaluated bys -1 br (n) (r) g (s) + F r g() – g( + n) r! =0 r =f G (n) = limsm.(172)When (s )N is really a periodic sequence of period p, due to the fact as outlined by Equation (170) every -1 -1 function of sort F r n=0 s z() is usually decomposed into a finite sum of terms F r n=0 ei z(), then f G (n) is the one of a kind all-natural expansion for f (n) presented in Equation (140), and for all n C, the function f G (n) could be written as f G (n) = eiss -1 r ( n ) (r ) g (s) + F r ei g() – ei (+n) g(.
