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It is known that the sum $\Phi_k(M)=\displaystyle\sum_{n=0}^M(-1)^nn^k$ can be written by Euler polynomials and $F_k(N)$ can be written by Bernoulli polynomials.

Historically , Euler on page 499 of "L.Euler, Institutiones Calculi Differentialis, Petersberg,1755", introduced Euler polynomials, to evaluate the alternating sum $\Phi_k(M)$

More precisely,

$$\sum_{i=0}^{n-1}i^p=\frac{1}{p+1}\sum_{k=0}^p\binom{p+1}{k}B_kn^{p+1-k}$$

where $B_k$ are Bernoulli numbers and can be defined by following generating function

$$\frac{x}{e^x-1}=\sum_{k=0}^\infty B_k\frac{x^k}{k!}$$

See this paper for irreducibility for such polynomials

Duke Math. J. Volume 19, Number 3 (1952), 475-481. Note on irreducibility of the Bernoulli and Euler polynomials

L. Carlitz

http://projecteuclid.org/euclid.dmj/1077477373

It is known that the sum $\Phi_k(M)=\displaystyle\sum_{n=0}^M(-1)^nn^k$ can be written by Euler polynomials and $F_k(N)$ can be written by Bernoulli polynomials.

Historically , Euler on page 499 of "L.Euler, Institutiones Calculi Differentialis, Petersberg,1755" (Euler archive link), introduced Euler polynomials, to evaluate the alternating sum $\Phi_k(M)$

More precisely,

$$\sum_{i=0}^{n-1}i^p=\frac{1}{p+1}\sum_{k=0}^p\binom{p+1}{k}B_kn^{p+1-k}$$

where $B_k$ are Bernoulli numbers and can be defined by following generating function

$$\frac{x}{e^x-1}=\sum_{k=0}^\infty B_k\frac{x^k}{k!}$$

See this paper for irreducibility for such polynomials

Duke Math. J. Volume 19, Number 3 (1952), 475-481. Note on irreducibility of the Bernoulli and Euler polynomials

L. Carlitz

http://projecteuclid.org/euclid.dmj/1077477373

$\Phi_k(M)=\displaystyle\sum_{n=0}^M(-1)^nn^k$ can be written by Euler polynomials and $F_k(N)$ can be written by Bernoulli polynomials.

More precisely,

$$\sum_{i=0}^{n-1}i^p=\frac{1}{p+1}\sum_{k=0}^p\binom{p+1}{k}B_kn^{p+1-k}$$

where $B_k$ are Bernoulli numbers and can be defined by following generating function

$$\frac{x}{e^x-1}=\sum_{k=0}^\infty B_k\frac{x^k}{k!}$$

See this paper for irreducibility for such polynomials

Duke Math. J. Volume 19, Number 3 (1952), 475-481. Note on irreducibility of the Bernoulli and Euler polynomials

L. Carlitz

http://projecteuclid.org/euclid.dmj/1077477373

It is known that the sum $\Phi_k(M)=\displaystyle\sum_{n=0}^M(-1)^nn^k$ can be written by Euler polynomials and $F_k(N)$ can be written by Bernoulli polynomials.

Historically , Euler on page 499 of "L.Euler, Institutiones Calculi Differentialis, Petersberg,1755", introduced Euler polynomials, to evaluate the alternating sum $\Phi_k(M)$

More precisely,

$$\sum_{i=0}^{n-1}i^p=\frac{1}{p+1}\sum_{k=0}^p\binom{p+1}{k}B_kn^{p+1-k}$$

where $B_k$ are Bernoulli numbers and can be defined by following generating function

$$\frac{x}{e^x-1}=\sum_{k=0}^\infty B_k\frac{x^k}{k!}$$

See this paper for irreducibility for such polynomials

Duke Math. J. Volume 19, Number 3 (1952), 475-481. Note on irreducibility of the Bernoulli and Euler polynomials

L. Carlitz

http://projecteuclid.org/euclid.dmj/1077477373

$\Phi_k(M)=\displaystyle\sum_{n=0}^M(-1)^nn^k$ can be written by Euler polynomials and $F_k(N)$ can be written by Bernoulli polynomials. See this paper

Duke Math. J. Volume 19, Number 3 (1952), 475-481. Note on irreducibility of the Bernoulli and Euler polynomials

L. Carlitz

http://projecteuclid.org/euclid.dmj/1077477373

$\Phi_k(M)=\displaystyle\sum_{n=0}^M(-1)^nn^k$ can be written by Euler polynomials and $F_k(N)$ can be written by Bernoulli polynomials.

More precisely,

$$\sum_{i=0}^{n-1}i^p=\frac{1}{p+1}\sum_{k=0}^p\binom{p+1}{k}B_kn^{p+1-k}$$

where $B_k$ are Bernoulli numbers and can be defined by following generating function

$$\frac{x}{e^x-1}=\sum_{k=0}^\infty B_k\frac{x^k}{k!}$$

See this paper for irreducibility for such polynomials

Duke Math. J. Volume 19, Number 3 (1952), 475-481. Note on irreducibility of the Bernoulli and Euler polynomials

L. Carlitz

http://projecteuclid.org/euclid.dmj/1077477373