{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T03:25:45Z","timestamp":1780370745337,"version":"3.54.1"},"reference-count":31,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2019,10,11]],"date-time":"2019-10-11T00:00:00Z","timestamp":1570752000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The aim of this paper is to construct generating functions for new families of combinatorial numbers and polynomials. By using these generating functions with their functional and differential equations, we not only investigate properties of these new families, but also derive many new identities, relations, derivative formulas, and combinatorial sums with the inclusion of binomials coefficients, falling factorial, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), the Poisson\u2013Charlier polynomials, combinatorial numbers and polynomials, the Bersntein basis functions, and the probability distribution functions. Furthermore, by applying the p-adic integrals and Riemann integral, we obtain some combinatorial sums including the binomial coefficients, falling factorial, the Bernoulli numbers, the Euler numbers, the Stirling numbers, the Bell polynomials (i.e., exponential polynomials), and the Cauchy numbers (or the Bernoulli numbers of the second kind). Finally, we give some remarks and observations on our results related to some probability distributions such as the binomial distribution and the Poisson distribution.<\/jats:p>","DOI":"10.3390\/axioms8040112","type":"journal-article","created":{"date-parts":[[2019,10,11]],"date-time":"2019-10-11T10:53:03Z","timestamp":1570791183000},"page":"112","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Generating Functions for New Families of Combinatorial Numbers and Polynomials: Approach to Poisson\u2013Charlier Polynomials and Probability Distribution Function"],"prefix":"10.3390","volume":"8","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-9100-2252","authenticated-orcid":false,"given":"Irem","family":"Kucukoglu","sequence":"first","affiliation":[{"name":"Department of Engineering Fundamental Sciences, Faculty of Engineering, Alanya Alaaddin Keykubat University, TR-07425 Antalya, Turkey"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2857-6629","authenticated-orcid":false,"given":"Burcin","family":"Simsek","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of Pittsburgh, Pittsburgh, PA 15260, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0611-7141","authenticated-orcid":false,"given":"Yilmaz","family":"Simsek","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science University of Akdeniz, TR-07058 Antalya, Turkey"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2019,10,11]]},"reference":[{"key":"ref_1","first-page":"1141","article-title":"On generating function of the Bernstein polynomials","volume":"CP1281","author":"Acikgoz","year":"2010","journal-title":"Proc. Int. Conf. Numer. Anal. Appl. Math. Am. Inst. Phys. Conf. Proc."},{"key":"ref_2","unstructured":"Bona, M. (2007). Introduction to Enumerative Combinatorics, The McGraw-Hill Companies Inc."},{"key":"ref_3","unstructured":"Charalambides, C.A. (2002). Enumerative Combinatorics, Chapman and Hall (CRC Press Company)."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Comtet, L. (1974). Advanced Combinatorics: The Art of Finite and Infinite Expansions, D. Reidel Publishing Company.","DOI":"10.1007\/978-94-010-2196-8"},{"key":"ref_5","unstructured":"Djordjevic, G.B., and Milovanovi\u0107, G.V. (2014). Special Classes of Polynomials, Faculty of Technology, University of Nis."},{"key":"ref_6","first-page":"126","article-title":"Aufgabe 1088","volume":"49","author":"Golombek","year":"1994","journal-title":"El. Math."},{"key":"ref_7","unstructured":"Jordan, C. (1950). Calculus of Finite Differences, Chelsea Publishing Company. [2nd ed.]."},{"key":"ref_8","first-page":"5969","article-title":"Daehee numbers and polynomials","volume":"7","author":"Kim","year":"2013","journal-title":"Appl. Math. Sci. (Ruse)"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"993","DOI":"10.12988\/astp.2013.39117","article-title":"A note on Changhee numbers and polynomials","volume":"7","author":"Kim","year":"2013","journal-title":"Adv. Stud. Theor. Phys."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"2166","DOI":"10.1016\/j.jnt.2008.11.004","article-title":"On Euler numbers, polynomials and related p-adic integrals","volume":"129","author":"Kim","year":"2009","journal-title":"J. Number Theory"},{"key":"ref_11","first-page":"288","article-title":"q-Volkenborn integration","volume":"19","author":"Kim","year":"2002","journal-title":"Russ. J. Math. Phys."},{"key":"ref_12","first-page":"7","article-title":"q-Euler numbers and polynomials associated with p-adic q-integral and basic q-zeta function","volume":"9","author":"Kim","year":"2006","journal-title":"Trend Math. Inf. Cent. Math. Sci."},{"key":"ref_13","first-page":"41","article-title":"Observations on identities and relations for interpolation functions and special numbers","volume":"28","author":"Kucukoglu","year":"2018","journal-title":"Adv. Stud. Contemp. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"2337","DOI":"10.3906\/mat-1906-6","article-title":"An approach to negative hypergeometric distribution by generating function for special numbers and polynomials","volume":"43","author":"Kucukoglu","year":"2019","journal-title":"Turk. J. Math."},{"key":"ref_15","unstructured":"Lorentz, G.G. (1986). Bernstein Polynomials, Chelsea Publishing Company."},{"key":"ref_16","first-page":"457","article-title":"On the Poisson\u2013Charlier polynomials","volume":"41","author":"Ozmen","year":"2015","journal-title":"Serdica Math. J."},{"key":"ref_17","doi-asserted-by":"crossref","unstructured":"Schikhof, W.H. (1984). Ultrametric Calculus: An Introduction to p-Adic Analysis, Cambridge University Press. Cambridge Studies in Advanced Mathematics 4.","DOI":"10.1017\/CBO9780511623844"},{"key":"ref_18","unstructured":"Roman, S. (2005). The Umbral Calculus, Dover Publishing Inc."},{"key":"ref_19","unstructured":"Ross, S.M. (2010). A First Course in Probability, Pearson Education, Inc.. [8th ed.]."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1186\/1687-1812-2013-80","article-title":"Functional equations from generating functions: A novel approach to deriving identities for the Bernstein basis functions","volume":"2013","author":"Simsek","year":"2013","journal-title":"Fixed Point Theory Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"343","DOI":"10.1186\/1687-1812-2013-87","article-title":"Generating functions for generalized Stirling type numbers, array type polynomials, Eulerian type polynomials and their applications","volume":"2013","author":"Simsek","year":"2013","journal-title":"Fixed Point Theory Appl."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2298\/AADM1801001S","article-title":"New families of special numbers for computing negative order Euler numbers and related numbers and polynomials","volume":"12","author":"Simsek","year":"2018","journal-title":"Appl. Anal. Discr. Math."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"557","DOI":"10.3906\/mat-1703-114","article-title":"Construction of Some New Families of Apostol-type Numbers and Polynomials via Dirichlet Character and p-adic q-integrals","volume":"42","author":"Simsek","year":"2018","journal-title":"Turk. J. Math."},{"key":"ref_24","first-page":"931","article-title":"Formulas for Poisson\u2013Charlier, Hermite, Milne-Thomson and other type polynomials by their generating functions and p-adic integral approach","volume":"113","author":"Simsek","year":"2019","journal-title":"RACSAM Rev. R. Acad. A"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"1328","DOI":"10.1016\/j.jmaa.2019.05.015","article-title":"Generating functions for finite sums involving higher powers of binomial coefficients: Analysis of hypergeometric functions including new families of polynomials and numbers","volume":"477","author":"Simsek","year":"2019","journal-title":"J. Math. Anal. Appl."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Simsek, Y., and Acikgoz, M. (2010). A new generating function of (q-) Bernstein-type polynomials and their interpolation function. Abstr. Appl. Anal., 769095.","DOI":"10.1155\/2010\/769095"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1017\/S0305004100004412","article-title":"Some formulas for the Bernoulli and Euler polynomials at rational arguments","volume":"129","author":"Srivastava","year":"2000","journal-title":"Math. Proc. Camb. Philos. Soc."},{"key":"ref_28","first-page":"390","article-title":"Some generalizations and basic (or q-) extensions of the Bernoulli, Euler and Genocchi polynomials","volume":"5","author":"Srivastava","year":"2011","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Srivastava, H.M., and Choi, J. (2001). Series Associated with the Zeta and Related Functions, Kluwer Acedemic Publishers.","DOI":"10.1007\/978-94-015-9672-5"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Srivastava, H.M., and Choi, J. (2012). Zeta and q-Zeta Functions and Associated Series and Integrals, Elsevier Science Publishers.","DOI":"10.1016\/B978-0-12-385218-2.00002-5"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"61","DOI":"10.2298\/AADM180804021X","article-title":"On an open problem of Simsek concerning the computation of a family special numbers","volume":"13","author":"Xu","year":"2019","journal-title":"Appl. Anal. Discret. 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