Novel Formulas for B-Splines, Bernstein Basis Functions and special numbers: Approach to Derivative and Functional Equations of Generating Functions

One of the main purposes of this article is to give functional equations and differential equations between Bernstein basis functions and generating functions of B-spline curves. Using these equations, very useful formulas containing the relationships among the uniform B-spline curves, the Bernstein basis functions, and other special numbers and polynomials are derived. By applying p-adic integrals to these polynomials, many novel formulas are also derived. Furthermore, by applying the partial differential derivative operator and Laplace transformation to these generating functions, with aid of higher derivative differential, not only recurrence relations and the higher derivative formula for B-spline curves, but also infinite series representations are given.


Introduction
Spline theory has been among the most popular areas of mathematics and other applied sciences in recent years.Isaac Jacob Schoenberg [29], a Romanian-American mathematician, is credited with the invention of splines and is also known as their father.The best known of these is B-spline.Generally, B-splines and the Bezier curves expressed in terms of the Bernstein basis functions are also known to be used for curve fitting and numerical differentiation of experimental data.Moreover, they are often used effectively in computer-aided design and computer graphics (cf.[8,10,7,18,17,28]; see also the references cited in each of these earlier works).The generating function for the B-spline was firstly constructed by Goldman [8].The motivation of this article is to give new formulas and finite sums that include B-splines and special polynomials by blending the generating functions with their functional equations for the Bernstein basis functions, the B-spline other special numbers and polynomials involving special functions, the Apostol type Bernoulli numbers and polynomials, the Apostol type Euler numbers and polynomials, the Eulerian numbers and polynomials, the Stirling numbers.
The remaining parts of this article the following notations are used: Let N, Z, Q, R and C demonstrate the set of natural numbers, the set of integers, the set of rational numbers, the set of real numbers and the set of complex numbers, respectively.N 0 = N∪ {0}.Let C p be the field of p-adic completion of algebraic closure of Q p , set of p-adic rational numbers.Let Z p be set of p-adic integers.
The remaining parts of this article are also briefly summarized as follows: In preliminaries section, we give generating functions with their some properties of special numbers and polynomials.
In Section 2, by using generating functions with their functional equations and using p-adic integrals, we give some novel computation formulas of the Apostol-Bernoulli polynomials, the Apostol-Euler polynomials, the Eulerian numbers and polynomials, and the Euler Frobenius numbers and polynomials.
In Section 3, with the aid of generating functions with their derivative and functional equations, we give many novel identities and relations for the uniform B-splines and the Bernstein basis functions.With the aid of application of Laplace transform to generating functions for the uniform B-splines, series representations of the uniform B-spline and the Bernstein basis functions are given.
Finally, this article is concluded with the conclusion section.
Generating functions for the Bernstein basis functions B k j (ω) are given by where (cf. [31,33]).In recent years, many articles have been published covering the generating functions of Bernstein base functions and their applications in many different fields.Some of these studies are (cf.[1,17,10,26,27,31]).The Bernstein polynomials and their properties are available in [18] and [28] in a very efficient and practical way.
By using (19), using generating function method, Goldman [8, Theorem 3] proved the following well-known Schoenberg's identity and de Boor recurrence for the uniform B-splines, respectively: where p ≤ ω ≤ p + 1 and By using (20), we values of the basis N 0,n (ω; p) are given as follows: For n = 2, we have Thus we have N 0,n (ω; p) = 0, p ≤ ω ≤ p + 1 if p ≥ n + 1 and so on.
A very brief introduction to the p-adic integrals are presented as follows: Let Ψ : Z p → C p be a uniformly differential function on Z p .The Volkenborn integral or the p-adic bosonic integral is given by where [25,37,13,16,35]; see also the references cited in each of these earlier works).

Computation Formulas of the Apostol-Bernoulli Polynomials, Eulerian Numbers and Polynomials
We give some computation formulas of the Apostol-Bernoulli polynomials.These formulas involving the Stirling numbers of the second kind, the Bernstein polynomials and the array polynomials.By applying p-adic integrals to these formulas, and using the Witt identities for the Bernoulli and Euler numbers, we give some identities and certain family of finite sums.By combining (3) with ( 5), we have Thus, the coefficients of t n n! on both sides of the previous equation are equalized yields (6).
Joining ( 6) with ( 4) and ( 18), for n ∈ N 0 , we have the following known result: Using ( 5), we also get Combining ( 24) with (2), we get Comparing the coefficients of t m m! on both sides of the last equation, we arrive at the following theorem: Combining ( 24) with (1), we get Comparing the coefficients of t m m! on both sides of the last equation, we arrive at the following theorem: By using (11), we get Combining the above equation with following generating function [34, Eq. ( 8)]), we get where assuming that ρ 1−ρ < 1, we have Comparing the coefficients of t m m! on both sides of the above equation, we arrive at the following theorem: where ρ 1−ρ < 1. Since (cf. [34]), ( 25) reduces to the following corollary: Since (cf. [9,34]), we get the following result: where ρ 1−ρ < 1.
Using (13), we get By combining ( 26) and ( 1), we get Comparing the coefficients of t m m! on both sides of the above equation yields Let m ∈ N. Then we have Combining ( 27) with (18), we arrive at the following corollary: By combining ( 9) and ( 13), we have the following functional equation: Using this equation, we get for n > 0.
By combining (14), and (4) or using (28), we give some values of the poly-nomials A n (ρ) as follows: and so on.Boyadzhiev [4] gave a relation between the Apostol-Bernoulli numbers and the geometric polynomials W n (w) is given as follows: and where n ∈ N and With the Euler (derivative) operator ρ d dρ , Boyadzhiev also showed that where |ρ| < 1. Joining the above equation with ( 14), we get the following result: Combining the above equation with (28) yields Joining the above equation with the following well-known equation which is combined ( 9) with ( 5): (cf. [36]), we also get We think that there are many different proofs of the equations ( 30) and (32).Some of them were also given by (cf.[3,4,11,19,20,21,36]).Combining ( 28) with ( 6) and ( 7) where n ∈ N. By applying Volkenborn integral ( 21) to (33) and using which is known as the Witt identity for the Bernoulli numbers, we obtain n j=1 (−1) j+1 n j j (ρ − 1) Combining the above equation with we arrive at the following theorem: By applying the Ferminoic p-adic integral ( 22) to (33) and using Zp which is known as the Witt identity for the Euler numbers, we get Comparing the above equation with we arrive at the following theorem: 3 Identities, Relations and Series Representations for the Uniform B-Splines and the Bernstein Basis Functions In this section, we give many new formulas involving uniform B-spline, Apostol-Bernoulli numbers and Eulerian numbers.We also give some functional equations and derivative equations for generating functions for the uniform B-splines and the Bernstein basis functions.Using these equations and the Laplace transform, we study series representations for the uniform B-splines and the Bernstein basis functions.

Relations among the uniform B-spline, Apostol-Bernoulli numbers and Eulerian numbers
Here, by using p-adic integrals, we introduce very interesting results among B-Spline, the Apostol-Bernoulli numbers and the Eulerian numbers.Let's briefly give these interesting relationships as follows: Combining ( 28) and ( 20) with ( 14) and (15), we arrive at the following result: A relation between the uniform B-Spline and the Eulerian numbers is also given by

Relations between the uniform B-spline and the Bersntein basis functions
By using generating functions and their functional equations and PDEs, relations between the uniform B-spline and the Bersntein basis functions are given.
where y > 0. By applying the Laplace transform to the above functional equation, we get Since the rest of proof of this theorem is similar to the that of (42), we skip this proof here.

Conclusions
In this article, the results and how they were obtained are discussed together with their methods.For this purpose, by applying p-adic integrals to the identities found for special polynomials, new formulas were given that maybe serve as a resource for researchers on the subject.In addition, by using the differential and functional equations of the generating functions, some novel formulas for special numbers and special polynomials were found.In addition, thanks to these methods, some formulas were given for finite sums containing these special numbers and polynomials.Derivative formulas of the B-spline curves were found by using the higher order derivative formula of Benstein base functions.Moreover, by applying the Laplace transform to the new generating functional equations we found, infinite series containing B-spline curves are given.
It is planned in the near future to develop new mathematical models and different applications using functional balances and differential equations of the generator functions obtained by blending B-spline curves and Bernstein basis functions.