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$$\sum _{x\ge0}^\Re f(x)=-\int_0^1 f(t) \,dt -\sum_{k=1}^\infty \frac{c_k \, \Delta^{k-1}f(1)}{k!}$$

where $c_k=\int _0^1 \frac {\Gamma (x+1)}{\Gamma (x-k+1)} \, dx$

$$\sum _{x\ge0}^\Re f(x)=\sum _{k=1}^\infty \frac {\Delta^{k-1}f(1)}{k!} (-1)_k$$

where $(x)_k=\lim_{s\to k} \frac {\Gamma (x+1)}{\Gamma (x-s+1)}$

$$\sum _{x\ge0}^\Re f(x)=-\sum _{n=1}^\infty \frac {f^{(n-1)}(0)}{n!} B_n(1)$$

$$\sum _{x\ge0}^\Re f(x)=-\int_0^1 f(t) \, dt + \frac {1}{2} f(1) - \sum_{k=1}^\infty \frac {B_{2k}}{(2k)!} f^{(2k-1)}(1)$$

Are all these definitions equal? If not, in what cases they are?

Are they equal if $f(x)$ is equal to its Newton series expansion?

$$\sum _{x\ge0}^\Re f(x)= -\sum_{k=1}^\infty \frac{c_k \, \Delta^{k-1}f(0)}{k!}$$

where $c_k=\int _0^1 \frac {\Gamma (x+1)}{\Gamma (x-k+1)} \, dx$

$$\sum _{x\ge0}^\Re f(x)=\sum _{k=1}^\infty \frac {\Delta^{k-1}f(0)}{k!} (0)_k$$

where $(x)_k=\lim_{s\to k} \frac {\Gamma (x+1)}{\Gamma (x-s+1)}$

$$\sum _{x\ge0}^\Re f(x)=-\sum _{n=1}^\infty \frac {f^{(n-1)}(0)}{n!} B_n(0)$$

$$\sum _{x\ge0}^\Re f(x)= \frac {1}{2} f(0) - \sum_{k=1}^\infty \frac {B_{2k}}{(2k)!} f^{(2k-1)}(0)$$

Are all these definitions equal? If not, in what cases they are?

Are they equal if $f(x)$ is equal to its Newton series expansion?

$$\sum _{x\ge0}^\Re f(x)=\int_0^x f(t) \,dt +\sum_{k=1}^\infty \frac{c_k \, \Delta^{k-1}f(x)}{k!}$$

where $c_k=\int _0^1 \frac {\Gamma (x+1)}{\Gamma (x-k+1)} \, dx$

$$\sum _{x\ge0}^\Re f(x)=-\sum _{k=1}^\infty \frac {\Delta^{k-1}f(x)}{k!} (-x)_k$$

where $(x)_k=\lim_{s\to k} \frac {\Gamma (x+1)}{\Gamma (x-s+1)}$

$$\sum _{x\ge0}^\Re f(x)=\sum _{n=1}^\infty \frac {f^{(n-1)}(0)}{n!} B_n(x)$$

$$\sum _{x\ge0}^\Re f(x)=\int_0^x f(t) \, dt - \frac {1}{2} f(x) + \sum_{k=1}^\infty \frac {B_{2k}}{(2k)!} f^{(2k-1)}(x)$$

Are all these definitions equal? If not, in what cases they are?

Are they equal if $f(x)$ is equal to its Newton series expansion?

$$\sum _{x\ge0}^\Re f(x)=-\int_0^1 f(t) \,dt -\sum_{k=1}^\infty \frac{c_k \, \Delta^{k-1}f(1)}{k!}$$

where $c_k=\int _0^1 \frac {\Gamma (x+1)}{\Gamma (x-k+1)} \, dx$

$$\sum _{x\ge0}^\Re f(x)=\sum _{k=1}^\infty \frac {\Delta^{k-1}f(1)}{k!} (-1)_k$$

where $(x)_k=\lim_{s\to k} \frac {\Gamma (x+1)}{\Gamma (x-s+1)}$

$$\sum _{x\ge0}^\Re f(x)=-\sum _{n=1}^\infty \frac {f^{(n-1)}(0)}{n!} B_n(1)$$

$$\sum _{x\ge0}^\Re f(x)=-\int_0^1 f(t) \, dt + \frac {1}{2} f(1) - \sum_{k=1}^\infty \frac {B_{2k}}{(2k)!} f^{(2k-1)}(1)$$

Are all these definitions equal? If not, in what cases they are?

Are they equal if $f(x)$ is equal to its Newton series expansion?

$$\sum _{x\ge0}^{\Re } f(x)=\int_0^x f(t) dt +\sum_{k=1}^\infty \frac{c_k\Delta^{k-1}f(x)}{k!}$$

where $c_{k}=\int _{0}^{1}{\frac {\Gamma (x+1)}{\Gamma (x-k+1)}}dx$

$$\sum _{x\ge0}^{\Re } f(x)=-\sum _{k=1}^{\infty }{\frac {\Delta ^{k-1}f(x)}{k!}}(-x)_{k}$$

where $(x)_{k}=\lim_{s\to k}{\frac {\Gamma (x+1)}{\Gamma (x-s+1)}}$

$$\sum _{x\ge0}^{\Re } f(x)=\sum _{n=1}^{\infty }{\frac {f^{(n-1)}(0)}{n!}}B_{n}(x)$$

$$\sum _{x\ge0}^{\Re } f(x)=\int _{0}^{x}f(t)dt-{\frac {1}{2}}f(x)+\sum _{k=1}^{\infty }{\frac {B_{2k}}{(2k)!}}f^{(2k-1)}(x)$$

Are all these definitions equal? If not, in what cases they are?

Are they equal if $f(x)$ is equal to its Newton series expansion?

$$\sum _{x\ge0}^\Re f(x)=\int_0^x f(t) \,dt +\sum_{k=1}^\infty \frac{c_k \, \Delta^{k-1}f(x)}{k!}$$

where $c_k=\int _0^1 \frac {\Gamma (x+1)}{\Gamma (x-k+1)} \, dx$

$$\sum _{x\ge0}^\Re f(x)=-\sum _{k=1}^\infty \frac {\Delta^{k-1}f(x)}{k!} (-x)_k$$

where $(x)_k=\lim_{s\to k} \frac {\Gamma (x+1)}{\Gamma (x-s+1)}$

$$\sum _{x\ge0}^\Re f(x)=\sum _{n=1}^\infty \frac {f^{(n-1)}(0)}{n!} B_n(x)$$

$$\sum _{x\ge0}^\Re f(x)=\int_0^x f(t) \, dt - \frac {1}{2} f(x) + \sum_{k=1}^\infty \frac {B_{2k}}{(2k)!} f^{(2k-1)}(x)$$

Are all these definitions equal? If not, in what cases they are?

Are they equal if $f(x)$ is equal to its Newton series expansion?