# Nörlund polynomials

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##### 2: 24.2 Definitions and Generating Functions
###### §24.2(ii) Euler Numbers and Polynomials
$(-1)^{n}E_{2n}>0$ .
##### 3: Bibliography C
• L. Carlitz (1960) Note on Nörlund’s polynomial $B_{n}^{(z)}$ . Proc. Amer. Math. Soc. 11 (3), pp. 452–455.
• ##### 4: 24.13 Integrals
###### §24.13(i) Bernoulli Polynomials
24.13.3 $\int_{x}^{x+(1/2)}B_{n}\left(t\right)\,\mathrm{d}t=\frac{E_{n}\left(2x\right)}% {2^{n+1}},$
For integrals of the form $\int_{0}^{x}B_{n}\left(t\right)B_{m}\left(t\right)\,\mathrm{d}t$ and $\int_{0}^{x}B_{n}\left(t\right)B_{m}\left(t\right)B_{k}\left(t\right)\,\mathrm% {d}t$ see Agoh and Dilcher (2011).
##### 5: 24.1 Special Notation
The notations $E_{n}$, $E_{n}\left(x\right)$, as defined in §24.2(ii), were used in Lucas (1891) and Nörlund (1924). …
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##### 9: 24.5 Recurrence Relations
###### §24.5 Recurrence Relations
24.5.1 $\sum_{k=0}^{n-1}{n\choose k}B_{k}\left(x\right)=nx^{n-1},$ $n=2,3,\dots$,
24.5.2 $\sum_{k=0}^{n}{n\choose k}E_{k}\left(x\right)+E_{n}\left(x\right)=2x^{n},$ $n=1,2,\dots$.
##### 10: 24.14 Sums
###### §24.14 Sums
24.14.5 $\sum_{k=0}^{n}{n\choose k}E_{k}\left(h\right)B_{n-k}\left(x\right)=2^{n}B_{n}% \left(\tfrac{1}{2}(x+h)\right),$