Abstract
The main objective of this article is to construct generating functions for central moments involving Bernstein basis functions. We give some alternating generating functions of these functions. We also give derivative formulas and a recurrence relation of central moments with the help of their generating functions. We also establish new relations between combinatorial numbers and polynomials, and also central moments. Furthermore, by applying Euler operator and Laplace transformation to central moments, we derive some important results. Finally, we give further remarks, observations and comments related to the content of this paper.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Bernstein based functions have many applications not only in mathematics but also in other sciences. These functions and their generating functions are elated to various areas such as B-splines, splines, probability generating functions and distribution functions, the Bezier curves, computer geometric designs, etc. For this reason, they still attract the attention of researchers working in many different fields. Therefore, the motivation of this article is to reveal new formulas and relations by blending generating functions for Bernstein basis functions, moment generating functions containing these basis functions, and some other special number families and polynomial families. These formulas and relations are related to the Apostol–Euler numbers and polynomials, the Bernstein basis functions, the Stirling numbers, the central moments, the combinatorial numbers and polynomials.
The Bernstein operator is defined as follows:
where \(f:\left[ 0,1\right] \rightarrow R\), \(n\in {\mathbb {N}}_{0}={\mathbb {N}} \cup \{0\}\), \({\mathbb {N}}=\left\{ {1,2,3,...}\right\} \), \(0\le j\le n\), and \(B_{j}^{n}(x)\) denotes the Bernstein basis functions:
which are defined by means of the following generating function:
With the aid of q analysis, generating function in (2) was found by Simsek and Acikgoz [22]. Equation (2) was also given Acikgoz and Arici [1, 15] and Simsek [15].
There are many applications of these generating functions, see also for detail (cf. [3, 6, 8, 16, 17]).
The Bernstein operator has remained popular for the following reasons: this operator is given explicitly for rational values of the variable x because derivatives and integrals of the Bernstein basis functions are easily calculated. These basis functions are related to many special functions, special polynomials, and also integral transform (cf. [2, 3, 9]; see also the references cited in each of these earlier works).
Many different mathematical models can be studied by extending Bernstein basis functions from the range \(0\le x\le 1\) to any range \(a\le x\le b\). Simsek [21] give a relation between generating functions for the uniform B-splines and generating functions for the Bernstein basis functions.
Substituting \(x=\frac{x-a}{b-a}\) into (1) and (2), we have
which are defined by means of the following generating functions:
(cf. [16]).
By using (3), pobability generating functions, moment generating functions and other applications were studied, see ( [6,7,8,9, 12,13,14, 16,17,20, 24]; see also the references cited in each of these earlier works).
Bernstein [2] gave the central moments involving the Bernstein basis function. This moment is defined by
where \(n\in {\mathbb {N}}\) and \(r\in {\mathbb {N}}_{0}\) (cf. [3, p. 106]).
There are many books, manuscripts, and papers for the the central moments given in (5) (cf. [2, 3, 9]; see also the references cited in each of these earlier works). Here we can use notation in [3]. There are also many other papers and books for the generating functions of the \(B_{k}^{n}(x)\). In [3, p. 106, Eq. (2.15)], the following polynomials are given as follows:
In [3], generating function for M(x; r, n) is given by the following theorem.
Theorem 1
Let \(n\in {\mathbb {N}}\) and \(0\le x\le n\). Then we have
Proof of (7) was firstly given by Bernstein [2]. The other proof given by same line by Bustamante [3].
The goal of this article is to give moments with aid of the Bernstein basis functions on the interval \(a\le x\le b\). Generating functions for these moments with their properties are given. Some functional equations with their applications of these functions are given. A polynomial T(x; r, n; a, b) is also defined. This polynomial can be used in generalized Voronovskaya-type theorems. In order to give these results, we need the Apostol–Euler polynomials of the first kind \({\mathcal {E}} _{n}(x,\lambda )\), which are defined by means of the following generating function:
Putting \(x=0\) in (8), generating function for Apostol–Euler numbers of the first kind is given as follows:
The Stirling numbers of the second kind are given by
where \(x_{(0)}=1\) and \(x_{(v)}=x(x-1)\cdots (x-v+1)\) (cf. [19, 23]).
Generating function for the following combinatorial numbers
are by
(cf. [18]).
The combinatorial numbers \(y_{4}(r,n;\tau ,u,w)\) are defined as follows:
(cf. [17]).
A brief summary of the sections of this article where the new results are exhibited is given as follows:
In Sect. 2, we give a new class of the polynomials T(x; r, n; a, b). These polynomials are also related to the Bernstein basis functions with real parameters and the central factorial moments. We construct generating functions for these polynomials with their properties. Using functional equations and derivative equations of these generating functions, we derive some formulas and relations.
In Sect. 3, by using partial derivative equations of the generating functions, we give higher-order derivative formulas for the polynomials T(x; t, n; a, b).
In Sect. 4, we introduce some integral formulas of the polynomials T(x; t, n; a, b). By applying the Laplace transformation to the generating functions, we derive infinite series representations for these polynomials and the Bernstein basis functions.
In Sect. 5, we give a conclusion section.
2 Generating functions for polynomials T(x; r, n; a, b)
In this section, we construct generating functions for the polynomials T(x; r, n; a, b), which involving the Bernstein basis functions with real parameters and the central factorial moments. Some properties of these functions can be given. We also give some identities for these polynomials with aid of functional equations of these generating functions.
Let a and b be any real parameters such that \(a\ne b\). Let \(a\le x\le b\).
Using (6) and (10), we give the following polynomial in terms of the Bernstein basis functions and the Stirling numbers of the second kind with real parameters a and b:
By using (14), we also have
Remark 1
Substituting \(a=0\) and \(b=1\), (15) reduces to (6).
Generating function for the polynomials T(x; r, n; a, b) is given by the following theorem.
Theorem 2
Let \(n\in {\mathbb {N}}\) and \(a\le x\le b\). Then we have
which yields
Proof
Joining the above equation with (14) yields
After some elementary calculations, we get
Therefore
Thus we arrive at the desired result. \(\square \)
Remark 2
By using (16), we get
This equation was proved by Simsek ([12, 13], and see also [14]).
Combining (16) with the equation (6) in [12], we get the following result:
Corollary 3
Let \(n\in {\mathbb {N}}\) and \(a\le x\le b\). Then we have
Joining (17), (11) with equation (13) in [12], we get
By applying the Cauchy product rule to the right-hand side of the above equation, we obtain the following theorem:
Theorem 4
Let \(n,r\in {\mathbb {N}}_{0}\). Then we have
Alternating generating functions for (16) are given as follows:
and
Combining (19) with (7), we get the following result:
Using the above functional equation, we get
Comparing coefficient \(\frac{t^{r}}{r!} \) on the both sides of the above equation with \(a=0 \) and \(b=1 \) yields (6).
By implementing (4) with (19 ), we get the following functional equation:
Using (20), we obtain the following functional equation:
By combining (21) with the following formula
where
and
which were given by Simsek [17], we get
Therefore,
After some calculations, we have
Comparing coefficient \(\frac{t^{r}}{r!}\) on the both sides of the above equation, we get the following theorem:
Theorem 5
Let \(n,k\in {\mathbb {N}}_{0}\) and \(a\le x\le b\). Then we have
Theorem 6
Let \(k\in {\mathbb {N}}_{0}\) and \(n\in {\mathbb {N}}\). Then we have
or
Proof
By applying the partial derivative operator \(\frac{\partial ^{k}}{\partial t^{k}}\) to both sides of the Eq. (16), we get
After some elementary calculations on the right side of the above equation, we complete proof of theorem. \(\square \)
Theorem 7
Let \(r\in {\mathbb {N}}\) and \(a\le x\le b\). Then we have
Proof
Taking derivative of (19) with respect to x, we get
Joining (22) with (8) and (9), we get the following partial derivative equation:
By implementing the above equation with (16), (8) and (9) and using the Cauchy product rule in the related series, we obtain
Comparing coefficient \(\frac{t^{r}}{r!}\) on the both sides of the above equation, we get
After some elementary calculations, we obtain
We now modify Theorem 11 in [4], we have
Combining the above equation with (23), we get the desired result. \(\square \)
Substituting \(a=0\) and \(b=1\) into (23), for \(T(x;r-1,n):=T(x;r-1,n;0,1),\) we also get the following corollary:
Corollary 8
Let \(r\in {\mathbb {N}}\) and \(0\le x\le 1\). Then we have
Taking derivative of (19) with respect to t, we get
Therefore
Using Cauchy product rule in the above equation yields
Comparing coefficient \(\frac{t^{r}}{r!}\) on the both sides of the above equation, we get the following theorem:
Theorem 9
(Recurrance relation) Let \(r\in {\mathbb {N}}\) and \(a\le x\le b\). Then we have
Combining (19) and (13), we get the following functional equation:
By using the above functional equation, we get
Comparing coefficient \(\frac{t^{r}}{r!}\) on the both sides of the above equation, we get the following theorem:
Theorem 10
Let \(r\in {\mathbb {N}}\) and \(a\le x\le b\). Then we have
3 Higher-order derivative formulas for T(x; r, n; a, b)
In this section, we give some novel higher-order derivative formulas for the polynomials T(x; r, n; a, b) with the aid of the partial derivative equations of \(\frac{\partial ^{l}}{\partial x^{l}}\left\{ {\mathcal {A}} (x;t,n;a,b)\right\} \) and \(\frac{\partial ^{l}}{\partial x^{l}}\left\{ B_{k}^{n}\left( x;a,b\right) \right\} \).
By applying Leibnitz’s derivative formula to (16) with respect to x, we get the following higher-order partial derivative equation:
By combining modified Theorem 3.8 in Simsek’s paper [15],
with (24), after some elementary calculations, we get
Therefore
Comparing coefficient \(\frac{t^{r}}{r!}\) on the both sides of the above equation, we get the following theorem:
Theorem 11
Let \(l\in {\mathbb {N}}_{0}\). Then we have
By taking \(l^{th}\) order derivative of (16) with respect to t, we get the following PDE:
By using the above equation, we get
Comparing coefficient \(\frac{t^{r}}{r!}\) on the both sides of the above equation, we get the following theorem:
Theorem 12
Let \(l,r,n\in {\mathbb {N}}_{0}\). Then we have
If we take \(l=1\), we get the following equation
Thus we get
By using the above equation, we get
Comparing coefficient \(\frac{t^{r}}{r!}\) on the both sides of the above equation, we get the following corollary:
Corollary 13
Let \(l,r,n\in {\mathbb {N}}_{0}\). Then we have
When \(a=0\) and \(b=1\), (25) reduces to following result:
Corollary 14
Let \(l,r,n\in {\mathbb {N}}_{0}\). Then we have
4 Integral formulas and application Laplace transformation to the generating functions for T(x; r, n; a, b)
In this section, we give some novel integral formulas for the polynomials T(x; r, n; a, b) with the aid of the Euler gamma function and beta function. By using these formulas, we derive some finite sums. By applying the Laplace transform to the generating function for the polynomials T(x; r, n; a, b), we also give infinite series representation for these polynomials and the Bernstein basis functions.
Therefore
After some calculations, we get
Other value of the integral (26) is also given by
Thus
Comparing (26) and (27) yields the following theorem:
Theorem 15
Let \(r,n\in {\mathbb {N}}_{0}\). Then we have
By using the integrals of
and
following corollary is obtained, respectively.
Corollary 16
Let \(r,n\in {\mathbb {N}}_{0}\). Then we have
and
By applying the Laplace transformation to the generating function of \({\mathcal {A}}(x;t,n;a,b)\), we obtain the following novel formula involving T(x; r, n; a, b) and \(B_{v}^{n}(x;a,b)\):
Theorem 17
Let \(n\in {\mathbb {N}}_{0}\). Then we have
Proof
From the Eq. (18), we get
Multiplying both sides of the above equation by \(e^{-t}\), we have
Integrate both sides of the above equation with respect to t from 0 to \( \infty \), we obtain
In order to guarantee the convergence of the above integral, we assume that \( t>0\) and \(t\left( n\left( \frac{x-a}{b-a}\right) +1-v\right) >0\), \( t,x\in {\mathbb {R}}\) and \(n,v\in {\mathbb {N}}\). By using these appropriate the conditions, the following Laplace transform of the function \(f(t)=t^{r}\) give us
on the left side of equation (29), we get
After some elementary calculations, we arrive the assertion of the theorem. \(\square \)
Substituting \(a=0\) and \(b=1\) into (28), we get the following result:
Corollary 18
Let \(r,n\in {\mathbb {N}}_{0}\). Then we have
Applications of the Laplace transform and other integral applications to the generating functions for the certain family of special polynomials were also given in ([4, 5, 11,12,13,14, 16,17,18, 20, 21]; see also the references cited in each of these earlier works).
5 Conclusion
The content of this paper was constructing generating functions for central moments which is involving Bernstein basis functions. This work is mainly geared to give derivative formulas and a recurrence relation of central moments with the help of their generating functions. Many formulas and identities were derived from these generating functions. Furthermore, we have showed relations between combinatorial polynomials and central moments. Not only some derivative and integral formulas for the polynomials T(x; r, n; a, b), but also some finite sums were given. By applying the Laplace transform to the generating function for the polynomials T(x; r, n; a, b), some infinite series representation for the polynomials T(x; r, n; a, b) and the Bernstein basis functions were given.
As we mentioned in the introduction, in addition to all the results discussed in this article, our results have a high potential to be used in applied sciences. This may provide us with new projects on the research and application of the Bernstein basis functions with their generating functions.
Data availability
There is no data in our paper.
References
Acikgoz, M., Araci, S.: On the generating function for Bernstein polynomials. American Institute of Physics Conference Proceedings CP1281, pp. 1141–1144 (2010)
Bernstein, S.N.: Demonstration of a theorem of Weierstrass based on the calculus of probabilities. Commun. Kharkov Math. Soc. 13, 1–2 (1912)
Bustamante, J.: Bernstein Operators and Their Properties. Birkhauser Springer International Publishing, Berlin (2017)
Gun, D., Simsek, Y.: Modification exponential Euler type splines derived from Apostol–Euler numbers and polynomials of complex order. Appl. Anal. Discrete Math. 17, 197–215 (2023)
Kilar, N., Simsek, Y., Srivastava, H.M.: Recurrence relations, associated formulas, and combinatorial sums for some parametrically generalized polynomials arising from an analysis of the Laplace transform and generating functions. Ramanujan J. 61, 731–756 (2023). https://doi.org/10.1007/s11139-022-00679-w
Kim, T., Choi, J., Kim, Y.H., Ryoo, C.S.: On the Fermionic \(p\)-adic integral representation of Bernstein polynomials associated with Euler numbers and polynomials. J. Inequal. Appl. 2010, 864247 (2010)
Kim, D., Simsek, Y., So, J.S.: Identities and computation formulas for combinatorial numbers including negative order Changhee polynomials. Symmetry 12(1), 9 (2020)
Kucukoglu, I., Simsek, B., Simsek, Y.: Multidimensional Bernstein polynomials and Bezier curves: analysis of machine learning algorithm for facial expression recognition based on curvature. Appl. Math. Comput. 344–345, 150–162 (2019)
Lorentz, G.G.: Bernstein Polynomials. Chelsea Publishing Company, New York (1986)
Luo, Q.M.: Apostol–Euler polynomials of higher order and Gaussian hypergeometric functions. Taiwan. J. Math. 10, 917–925 (2006)
Rainville, E.D.: Special Functions. The Macmillan Company, New York (1960)
Simsek, B.: Formulas derived from moment generating functions and Bernstein polynomials. Appl. Anal. Discrete Math. 13(3), 839–848 (2019)
Simsek, B.: A note on characteristic function for Bernstein polynomials involving special numbers and polynomials. Filomat 34(2), 543–549 (2020)
Simsek, B., Simsek, B.: The computation of expected values and moments of special polynomials via characteristic and generating functions. AIP Conf. Proc. 1863(1), 543–549 (2017)
Simsek, Y.: Functional equations from generating functions: a novel approach to deriving identities for the Bernstein basis functions. Fixed Point Theory Appl. 80, 1–13 (2013)
Simsek, Y.: Generating functions for the Bernstein type polynomials: a new approach to deriving identities and applications for the polynomials. Hacet. J. Math. Stat. 43(1), 1–14 (2014)
Simsek, Y.: Construction method for generating functions of special numbers and polynomials arising from analysis of new operators. Math. Methods Appl. Sci. 41, 6934–6954 (2018)
Simsek, Y.: New families of special numbers for computing negative order Euler numbers and related numbers and polynomials. Appl. Anal. Discrete Math. 12, 1–35 (2018)
Simsek, Y.: Explicit formulas for \(p\)-adic integral: approach to \(p\)-adic distributions and some families of special numbers and polynomials. Montes Taurus J. Pure Appl. Math. 1(1), 1–76 (2019)
Simsek, Y.: Formulas for Poisson–Charlier, Hermite, Milne–Thomson and other type polynomials by their generating functions and p-adic integral approach. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM 113, 931–948 (2019). https://doi.org/10.1007/s13398-018-0528-6
Simsek, Y.: Novel formulas for B-Splines, Bernstein basis functions, and special numbers: approach to derivative and functional equations of generating functions. Mathematics 12(1), 65 (2024). https://doi.org/10.3390/math12010065
Simsek, Y., Acikgoz, M.: A new generating function of (\(q\)-) Bernstein-type polynomials and their interpolation function. Abstr. Appl. Anal. 2010, 769095 (2010). https://doi.org/10.1155/2010/769095
Srivastava, H.M.: Zeta and \(q\)-Zeta Functions and Associated Series and Integrals. Elsevier, Amsterdam (2012)
Yalcin, F., Simsek, Y.: Formulas for characteristic function and moment generating functions of beta type distribution. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat. RACSAM (2022). https://doi.org/10.1007/s13398-022-01229-1
Funding
Open access funding provided by the Scientific and Technological Research Council of Türkiye (TÜBİTAK).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Ceylan, A.Y., Simsek, Y. Generating functions for polynomials derived from central moments involving bernstein basis functions and their applications. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat. 118, 57 (2024). https://doi.org/10.1007/s13398-024-01558-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13398-024-01558-3
Keywords
- Apostol–Euler numbers and polynomials
- Bernstein polynomials
- Functional equation
- Generating function
- Laplace transform
- Special functions
- Stirling numbers