SOME RELATIONSHIPS BETWEEN THE NUMBERS OF LYNDON WORDS AND A CERTAIN CLASS OF COMBINATORIAL NUMBERS CONTAINING POWERS OF BINOMIAL COEFFICIENTS

IREM KUCUKOGLU 장전수학회 2020 Advanced Studies in Contemporary Mathematics Vol.30 No.4

The main aim of this paper is to nd out some relationships between the numbers of Lyndon words and a certain class of combinatorial numbers which contains nite sums of powers of binomial coecients and whose generating functions were constructed and investigated by Simsek in \Generating functions for nite sums involving higher powers of binomial coecients: Analysis of hypergeometric functions including new families of polynomials and numbers, J. Math. Anal. Appl. 477 (2019), 1328{1352". By applying not only the Dirichlet convolution formula, but also the Mobius inversion formula, we obtain some identities containing the Mobius function, the Euler's totient function, the numbers of Lyndon words, the numbers of necklaces, the Stirling numbers of the second kind, and the aforementioned class of combinatorial numbers. Moreover, a few special cases and consequences of our results are considered. In particular, it should be noted here that in the special case when we take the power of the binomial coecients to be 1, some of our results are reduced to several results obtained by Kucukoglu and Simsek in \Identities and Derivative Formulas for the Combinatorial and Apostol-Euler Type Numbers by Their Generating Functions, Filomat 32(20) (2018), 6879{6891".

IDENTITIES AND RELATIONS ON THE q-APOSTOL TYPE FROBENIUS-EULER NUMBERS AND POLYNOMIALS

Kucukoglu, Irem,Simsek, Yilmaz Korean Mathematical Society 2019 대한수학회지 Vol.56 No.1

The main purpose of this paper is to investigate the q-Apostol type Frobenius-Euler numbers and polynomials. By using generating functions for these numbers and polynomials, we derive some alternative summation formulas including powers of consecutive q-integers. By using infinite series representation for q-Apostol type Frobenius-Euler numbers and polynomials including their interpolation functions, we not only give some identities and relations for these numbers and polynomials, but also define generating functions for new numbers and polynomials. Further we give remarks and observations on generating functions for these new numbers and polynomials. By using these generating functions, we derive recurrence relations and finite sums related to these numbers and polynomials. Moreover, by applying higher-order derivative to these generating functions, we derive some new formulas including the Hurwitz-Lerch zeta function, the Apostol-Bernoulli numbers and the Apostol-Euler numbers. Finally, for an application of the generating functions, we derive a multiplication formula, which is very important property in the theories of normalized polynomials and Dedekind type sums.

OBSERVATIONS ON IDENTITIES AND RELATIONS FOR INTERPOLATION FUNCTIONS AND SPECIAL NUMBERS

Irem Kucukoglu,Yilmaz Simsek 장전수학회 2018 Advanced Studies in Contemporary Mathematics Vol.28 No.1

The main motivation of this paper is to study and inves- tigate a new family of combinatorial numbers with their generating functions. Firstly, we obtain some finite series representations includ- ing well-known numbers such as the Apostol-Bernoulli numbers, the Apostol-Euler numbers, a family of combinatorial numbers, the Daehee numbers, the Changhee numbers and the Stirling numbers of the second kind. Secondly, applying Mellin transform to these functions, we give interpolation functions for these numbers. We investigate some proper- ties of these functions and other related complex valued functions. We observe that some special values of these functions give us the terms of some well-known infinite series. Thus, these functions unify the terms of some well-known identities and functions such as Hasse identity, the polylogarithm function, the digamma function, the Riemann zeta func- tions, the alternating Riemann zeta function, the Hurwitz zeta function, the alternating Hurwitz zeta function, the Hurwitz-Lerch zeta function and the other functions. Moreover, we give some remarks and observa- tions about these functions related to some special numbers and poly- nomials such as the Stirling numbers of the second kind, the harmonic numbers, the array polynomials and also related to hypergeometric func- tions, the family of zeta functions. We also give not only Riemann inte- gral representation, but also Cauchy integral representations for this new family of combinatorial numbers. Finally, in order to compute numerical values of these interpolation functions and other related complex valued functions, we present two algorithms. Furthermore, by using these al- gorithms, we provide some plots of these functions. Also, we investigate the effects of their parameters.

Identities and relations on the $q$-Apostol type Frobenius-Euler numbers and polynomials

Irem Kucukoglu,Yilmaz Simsek 대한수학회 2019 대한수학회지 Vol.56 No.1

The main purpose of this paper is to investigate the $q$-Apostol type Frobenius-Euler numbers and polynomials. By using generating functions for these numbers and polynomials, we derive some alternative summation formulas including powers of consecutive $q$-integers. By using infinite series representation for $q$-Apostol type Frobenius-Euler numbers and polynomials including their interpolation functions, we not only give some identities and relations for these numbers and polynomials, but also define generating functions for new numbers and polynomials. Further we give remarks and observations on generating functions for these new numbers and polynomials. By using these generating functions, we derive recurrence relations and finite sums related to these numbers and polynomials. Moreover, by applying higher-order derivative to these generating functions, we derive some new formulas including the Hurwitz--Lerch zeta function, the Apostol-Bernoulli numbers and the Apostol-Euler numbers. Finally, for an application of the generating functions, we derive a multiplication formula, which is very important property in the theories of normalized polynomials and Dedekind type sums.

REMARKS ON RECURRENCE FORMULAS FOR THE APOSTOL-TYPE NUMBERS AND POLYNOMIALS

IREM KUCUKOGLU,YILMAZ SIMSEK 장전수학회 2018 Advanced Studies in Contemporary Mathematics Vol.28 No.4

In this paper, by differentiating the generating functions for one of the family of the Apostol-type numbers and polynomials with re- spect to their parameters, we present some partial differential equations including these functions. By making use of these equations, we provide some new formulas, relations and identities including these numbers and polynomials and their derivatives. Furthermore, by using a col- lection of the generating functions for the aforementioned family and their functional equations, we investigate the numbers and polynomials belonging to this family and their relationships with other well-known special numbers and polynomials including the Apostol-Bernoulli num- bers and polynomials of higher order, the Apostol-Euler numbers and polynomials of higher order, the Frobenius-Euler numbers and polyno- mials of higher order, the λ-array polynomials, the λ-Stirling numbers, and the λ-Bernoulli numbers and polynomials.

A SURVEY ON SOME OLD AND NEW IDENTITIES ASSOCIATED WITH LAPLACE DISTRIBUTION AND BERNOULLI NUMBERS

ZEHRA SELIN ASKAN,IREM KUCUKOGLU,Yilmaz Simsek 장전수학회 2020 Advanced Studies in Contemporary Mathematics Vol.30 No.4

The purpose of this paper is to give some survey on old and new identities related to the characteristic function, the Laplace distribution and special numbers and polynomials with comparative results and observations. Additionally, we give some computation formulas for the higher-order moments of some kinds of random variables with the Laplace distribution in terms of the Bernoulli numbers of the first kind, the Euler numbers of the second kind and Riemann zeta function by using the techniques of generating functions and characteristic function of the aforementioned random variables. Finally, with the aid of the Hankel determinants formed by the moments corresponding to the weight function that reveals the orthogonality feature of the orthogonal polynomials, we give futher remarks and observations on not only orthogonality properties of some orthogonal polynomials such as the Hermite polynomials, but also construction methods of the three-term recurrence relations for the orthogonal polynomials.

Relationships between Skin Cancers and Blood Groups - Link between Non-melanomas and ABO/Rh Factors

Cihan, Yasemin Benderli,Baykan, Halit,Kavuncuoglu, Erhan,Mutlu, Hasan,Kucukoglu, Mehmet Burhan,Ozyurt, Kemal,Oguz, Arzu Asian Pacific Journal of Cancer Prevention 2013 Asian Pacific journal of cancer prevention Vol.14 No.7

Background: This investigation focused on possible relationships between skin cancers and ABO/Rh blood groups. Materials and Methods: Between January 2005 and December 2012, medical data of 255 patients with skin cancers who were admitted to Kayseri Training and Research Hospital, Radiation Oncology and Plastic Surgery Outpatient Clinics were retrospectively analyzed. Blood groups of these patients were recorded. The control group consisted of 25701 healthy volunteers who were admitted to Kayseri Training and Research Hospital, Blood Donation Center between January 2010 and December 2011. The distribution of the blood groups of the patients with skin cancers was compared to the distribution of ABO/Rh blood groups of healthy controls. The association of the histopathological subtypes of skin cancer with the blood groups was also investigated. Results: Of the patients, 50.2% had A type, 26.3% had O type, 16.1% had B type, and 7.5% had AB blood group with a positive Rh (+) in 77.3%. Of the controls, 44.3% had A type, 31.5% had 0 type, 16.1% had B type, and 8.1% had AB blood group with a positive Rh (+) in 87.8%. There was a statistically significant difference in the distribution of blood groups and Rh factors (A Rh (-) and 0 Rh positive) between the patients and controls. A total of 36.8% and 20.4% of the patients with basal cell carcinoma (BCC) had A Rh (+) and B Rh (+), respectively, while 39.2% and 27.6% of the controls had A Rh (+) and B Rh (+), respectively. A significant relationship was observed between the patients with BCC and controls in terms of A Rh (-) (p=0.001). Conclusion: Our study results demonstrated that there is a significant relationship between non-melanoma skin cancer and ABO/Rh factors.