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2, by using generating functions and their functional equations, we give some identities including the three-variable polynomials (y_{6} ( n;x,y,z a,b,v ) ), the Hermite polynomials, the array polynomials and the Stirling numbers of the second kind.
By using generating functions, we derive various functional equations and PDEs.
This study will give an opportunity to make a new classification of infinite groups and monoids by using generating functions.
Therefore, by using generating functions, to obtain a new test on the minimality of monoids would be an interesting and important result.
By using generating functions of q-Bernoulli, q-Euler numbers, and polynomials, we also present the q-analogues of ζ-type functions.
In this section, by using generating functions, we prove the Marsden identity for the unification of the Bernstein-type polynomials S n ( b, s, x ).
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In this paper, by using generating function of this numbers, we study on -Genocchi zeta and -functions.
Liu [5] discussed a class of strong deviation theorems for an arbitrary stochastic sequence with respect to the marginal distribution by using generating function method, and also studied the problem above by means of Laplace transform [6].
Loopless triangulations of a polygon with k vertices in k+2n triangles (with interior points and possibly multiple edges) were enumerated by Mullin in 1965, using generating functions and calculations with the quadratic method.
By using the generating functions, we obtain functional equations related to a generalized partial Hecke operator and Euler-type polynomials and numbers.
By using these generating functions, we derive many functional equations and differential equations.
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