Revision as of 10:55, 23 March 2015

The Euler polynomials $E_n(x)$ are orthogonal polynomials defined by $$E_n(x)=\displaystyle\sum_{k=0}^n {n \choose k} \dfrac{e_k}{2^k} \left( x - \dfrac{1}{2} \right)^{n-k},$$ where $e_k$ denotes an Euler number.

• $E_0(x)=1$
• $E_1(x)=x-\dfrac{1}{2}$
• $E_2(x)=x^2-x$
• $E_3(x)=x^3-\dfrac{3}{2}x^2+\dfrac{1}{4}$
• $E_4(x)=x^4-2x^3+x$

Properties

Theorem

The following formula holds for $|z|<\pi$: $$\dfrac{2e^{xt}}{e^t+1} = \sum_{k=0}^{\infty} \dfrac{E_n(x)t^n}{n!},$$ where $e^{xt}$ denotes the exponential function and $E_n$ denotes an Euler E polynomial.

Proof

References

• 1953: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: Higher Transcendental Functions Volume I ... (previous) ... (next): $\S 1.14 (2)$

Laguerre $L_n^{(\alpha)}$

Bateman $F_n$

Bernoulli $B$

Bessel poly

Chebyshev T

Chebyshev U

Dickson

Euler $E$

Gegenbauer $C$

Hermite (phys)

Hermite (prob)

Jacobi $P^{(\alpha,\beta)}$

Laguerre $L$

Legendre $P$

Bernoulli $B$

Chebyshev $T$

Fibonacci

Lucas