DLMF:2.10.E1 (Q988)

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DLMF:2.10.E1
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    Defining formula
    ∑ j = a n f ⁢ ( j ) = ∫ a n f ⁢ ( x ) ⁢ d x + 1 2 ⁢ f ⁢ ( a ) + 1 2 ⁢ f ⁢ ( n ) + ∑ s = 1 m - 1 B 2 ⁢ s ( 2 ⁢ s ) ! ⁢ ( f ( 2 ⁢ s - 1 ) ⁢ ( n ) - f ( 2 ⁢ s - 1 ) ⁢ ( a ) ) + ∫ a n B 2 ⁢ m - B ~ 2 ⁢ m ⁡ ( x ) ( 2 ⁢ m ) ! ⁢ f ( 2 ⁢ m ) ⁢ ( x ) ⁢ d x . superscript subscript 𝑗 𝑎 𝑛 𝑓 𝑗 superscript subscript 𝑎 𝑛 𝑓 𝑥 𝑥 1 2 𝑓 𝑎 1 2 𝑓 𝑛 superscript subscript 𝑠 1 𝑚 1 Bernoulli-number-B 2 𝑠 2 𝑠 superscript 𝑓 2 𝑠 1 𝑛 superscript 𝑓 2 𝑠 1 𝑎 superscript subscript 𝑎 𝑛 Bernoulli-number-B 2 𝑚 periodic-Bernoulli-polynomial-B 2 𝑚 𝑥 2 𝑚 superscript 𝑓 2 𝑚 𝑥 𝑥 {\displaystyle{\displaystyle\sum_{j=a}^{n}f(j)=\int_{a}^{n}f(x)\mathrm{d}x+% \tfrac{1}{2}f(a)+\tfrac{1}{2}f(n)+\sum_{s=1}^{m-1}\frac{B_{2s}}{(2s)!}\left(f^% {(2s-1)}(n)-f^{(2s-1)}(a)\right)+\int_{a}^{n}\frac{B_{2m}-\widetilde{B}_{2m}% \left(x\right)}{(2m)!}f^{(2m)}(x)\mathrm{d}x.}}
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    DLMF id
    DLMF:2.10.E1
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