Sentence examples for multiplication formula for from inspiring English sources

Several new results of independent interest are obtained as necessary steps in our analysis, in particular: (i) a multiplication formula for free Poisson multiple integrals, (ii) diagram formulae and spectral bounds for these objects, and (iii) a counterexample to the general universality of the Gaussian Wiener chaos in a classical setting.

This is the multiplication formula for Euler polynomials together with the relatively new identity mentioned in (4.19).

Remark 4.11 Kurt and Simsek [32] proved multiplication formula for the generalized Bernoulli polynomials of order α.

In this section we give a unified multiplication formula for the Apostol-type polynomials (mathcal{F}_{n}^{ (alpha ) } (x;lambda; mu nu )).

Remark 4.15 Walum [56] defined multiplication formula for periodic functions as follows: ϑ ( y ) f ( y x ) = ∑ j ( y ) f ( x + j y ), (36).

When a = λ = 1 and b = c = e into Theorem 4.12, we have the multiplication formula for the Bernoulli polynomials given by B n ( y x ) = y n − 1 ∑ j = 0 y − 1 B n ( x + j y ), (31).

Luo (Integral Transforms Spec. Funct. 20 377-391, 2009), introduced the lambda-multiple power sum and proved the multiplication formulas for the Apostol-Bernoulli and Apostol-Euler polynomials of higher order.

Luo in [6, 17] gave multiplication formulas for the Apostol-Bernoulli and Apostol-Euler polynomials.

All multiplication formulas for λ are then essentially special cases of addition formulas for Jacobian elliptic functions.

Remark 3.4 By substituting a = 1 and v = 0 into Theorem 3.1, Theorem 3.2 and Theorem 3.3, one can obtain multiplication formulas for the Bernoulli, Euler and Genocchi polynomials (cf. [1, 2, 4 16]).

The multiplication formulas for the Bernoulli and Euler polynomials are given as follows: ∑ k = 0 m − 1 B n ( x + k m ) = m 1 − n B n ( x ), (3).