By using functional equations of generating functions for special numbers and polynomials, we give some relations and identities including the Genocchi polynomials of negative order, the Euler numbers and polyn
By using the generating functions for the Bernstein basis functions, we derive various functional equations, differential equations and second order partial differential equations. By using these equations, we give new proofs of various identities, relations, integrals and derivatives of the Bernstein basi...
Algebra and Trigonometry: Structure and Method Book review THE BEST WORKSHEETS OF REFLECTION FOR ELEMENTRY SCHOOLS how do we use graphing linear inequalities in everyday life nonhomogeneous differential equation sofmath mcdougal littell chapter 2 answers calculation of logarithmic functions on ti...
The optimisation of an ansatz function is a flexible approach for working out a proper error mitigation formula. Various ansatz functions have been proposed36,37,38. In this work, we consider a general framework of this approach and focus on the scaling of the bias with respect to the gate ...
2. New general bounds for the ratio of gamma functions Alzer proved in his excellent paper [1, Theorem 8] that for every n≥0, the functions (9)Fn(x)=lnΓ(x)−(x−12)lnx+x−12ln2π−∑i=12nB2i2i(2i−1)x2i−1 and (10)Gn(x)=−lnΓ(x)+(x−12)lnx−x+12...
by using integral equations, we derive generating functions for new families of special numbers and polynomials. By using generating functions, we give relations between the Lah numbers, the Bernoulli numbers, the Euler numbers and the Laguerre polynomials. We also give further comments and remarks ...
Abstract The theory of theta functions is used to derive hypergeometric transformation formulas of degrees 3, 7, 11 and 23. As a consequence of the theory that is developed, some new series for 1/π are obtained that are similar to a class investigated by Ramanujan....
The three fundamental trigonometric ratios sinθ,cosθ and tanθ of an angle θ are very closely connected by relations. Thus, if any one of the ratios is known, the other two ratios can be easily calculated.In the trigonometric ratios, for any acute angle θ, we have...
functions, and we prepare a general result on estimates for certain convolution integrals. In Sect.3, we present and discuss the main results of the paper. The linearized time-periodic problem is addressed in Sect.4. Firstly, we deduce integral representation formulas for the velocity and ...
These susceptibilities will appear in the constitutive relations and in equilibrium correlation functions for fluids in flat space without external U(1) fields. As we will see later, the second-order thermodynamic transport coefficients in QCD at non-zero baryon number chemical potential (in flat ...