The Bernoulli polynomials of the second kind ψn(x), also known as the Fontana–Bessel polynomials, are the polynomials defined by the following generating...

In mathematics, the Bernoulli polynomials, named after Jacob Bernoulli, combine the Bernoulli numbers and binomial coefficients. They are used for series...

{C_{n}(n-1)!}{(v)_{n}}},\qquad \Re (v)>1,} A series with the Bernoulli polynomials of the second kind has the following form ψ ( v ) = ln ⁡ ( v + a ) + ∑ n = 1...

{B_{k+1+j}}{k+1+j}}={\frac {k!m!}{(k+m+1)!}}} Bernoulli polynomial Bernoulli polynomials of the second kind Bernoulli umbra Bell number Euler number Genocchi...

Bernoulli numbers of the second kind, or Cauchy numbers of the first kind, are the rational numbers that occur in the Maclaurin series expansion of the...

The Laguerre polynomials; The monomials ( xn : n = 0, 1, 2, ... ); The Mott polynomials; The Bernoulli polynomials of the second kind; The Falling and...

)}{\Big \}},\quad a>-1} where ψn(a) are the Bernoulli polynomials of the second kind, which are defined by the generating function z ( 1 + z ) s log ⁡...

the polynomials in a on the right-hand sides of these identities Faulhaber polynomials. These polynomials are divisible by a2 because the Bernoulli number...

functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions y(x) of Bessel's differential...

Because the Bernoulli polynomials may be written in terms of these forward differences, one immediately obtains a relation in the Bernoulli numbers:...

include the Newton polynomials, Selberg's polynomials, and the Stirling interpolation polynomials as special cases. The general difference polynomial sequence...

to the Stirling numbers, the Bernoulli numbers, and the generalized Bernoulli polynomials. There are multiple variants of the Stirling polynomial sequence...

involving the Stirling numbers hold for the Bernoulli polynomials. Many relations for the Stirling numbers shadow similar relations on the binomial coefficients...

the mathematical field of numerical analysis, a Bernstein polynomial is a polynomial expressed as a linear combination of Bernstein basis polynomials...

}}s^{n-2}g_{n}(s)} . Bernoulli polynomials of the second kind Stirling polynomials Poly-Bernoulli number see the formula section of OEIS A142978 see OEIS...

Euler–Bernoulli beam theory (also known as engineer's beam theory or classical beam theory) is a simplification of the linear theory of elasticity which...

about homogeneous polynomials. Euler polynomials Euler spline – splines composed of arcs using Euler polynomials Contributions of Leonhard Euler to mathematics...

numbers. Bernoulli numbers Stirling numbers Gregory coefficients Bernoulli polynomials Bernoulli polynomials of the second kind Stirling polynomials Khera...

Boyadzhiev, Khristo N. (2007). "Apostol-Bernoulli functions, derivative polynomials and Eulerian polynomials". arXiv:0710.1124 [math.CA]. Petersen, T...

non-constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since...

\quad \Re (a)>-1} where ψn(a) are the Bernoulli polynomials of the second kind and Nn,r(a) are the polynomials given by the generating equation ( 1 + z )...

sum is the limit of the infinite sequence of the Taylor polynomials. A function may differ from the sum of its Taylor series, even if its Taylor series...

{\displaystyle \cosh(t)} is the hyperbolic cosine function. The Euler numbers are related to a special value of the Euler polynomials, namely: E n = 2 n E n...

after John Napier. The Swiss mathematician Jacob Bernoulli discovered the constant while studying compound interest. The number e is of great importance...

of solutions. Jacob Bernoulli proposed the Bernoulli differential equation in 1695. This is an ordinary differential equation of the form y ′ + P ( x )...

domain of sufficiently differentiable functions. It is easy to check the symmetry as applied to monomials, so that one can take polynomials in the xi as...

numbers of the first kind Stirling numbers of the second kind Triangular number Triangular pyramidal number The (incomplete) Bell polynomials from a triangular...

of the Ehrhart polynomials, polynomials that count the number of integer points in a polygon or polyhedron when it is expanded by a given factor. The...

recursive formula for Chebyshev polynomials of the first kind. In the language of topology, Euler's formula states that the imaginary exponential function...

except the non-positive integers. For every positive integer n, Γ ( n ) = ( n − 1 ) ! . {\displaystyle \Gamma (n)=(n-1)!\,.} Derived by Daniel Bernoulli, for...