Many authors studied the q-**Daehee** **polynomials** (1.5), the degenerate λ-q-**Daehee** poly- nomials of the second kind in [12, 33, 46]. In this paper, we deﬁned the modiﬁed q-**Daehee** **polynomials** of the second kind (2.1), which are diﬀerent from the q-**Daehee** **polynomials** (1.5), and the modiﬁed degenerate q-**Daehee** **polynomials** of the second kind (3.1), which are diﬀerent from the modiﬁed q-**Daehee** numbers and **polynomials** in [31]. We obtained the interesting results of Theorems 2.1, 2.2, and 2.3, which are some identity properties related with the modiﬁed degenerate q-**Daehee** **polynomials** of the second kind (3.1) and also we obtained the results of Theorems 3.1, 3.2, and 3.3, which are some identities related with the modiﬁed q-**Daehee** **polynomials** of the second kind.

higher-order Bernoulli numbers and Bernoulli numbers of the second kind. In this paper, we give a p-adic integral representation of the twisted **Daehee** **polynomials** with q-parameter, and we derive some interesting properties related to the nth twisted **Daehee** **polynomials** with q-parameter.

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In this paper, we derive multifarious relationships among the two types of higher order q-**Daehee** **polynomials** and p-adic gamma function via Mahler theorem. Also, we compute some weighted p-adic q-integrals of the derivative of p-adic gamma function related to the Stirling numbers of the both kinds and the q-Bernoulli **polynomials** of order k.

function via their Mahler expansions. We also derived two q-Volkenborn integrals of p-adic gamma function in terms of q-**Daehee** **polynomials** and numbers and q-**Daehee** **polynomials** and numbers of the second kind. Moreover, we discover q-Volkenborn integral of the derivative of p-adic gamma function. We acquire the relationship between the p-adic gamma function and Stirling numbers of the …rst kind. We …nally develop a novel and interesting representation for the p-adic Euler constant by means of the q-**Daehee** **polynomials** and numbers.

In this paper, by considering Barnes-type **Daehee** **polynomials** of the ﬁrst kind as well as poly-Cauchy **polynomials** of the ﬁrst kind, we deﬁne and investigate the mixed-type poly- nomials of these **polynomials**. From the properties of Sheﬀer sequences of these polyno- mials arising from umbral calculus, we derive new and interesting identities.

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The totally degenerate **Daehee** numbers and **polynomials** are constructed by degener- ating both numerator and denominator of **Daehee** numbers and **polynomials**. From the generating function for the totally degenerate **Daehee** **polynomials** (), we can see that the totally degenerate **Daehee** **polynomials** are a Sheﬀer sequence. And we have the fol- lowing theorem.

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In this paper, we introduce the mixed-type **polynomials**: Barnes-type **Daehee** **polynomials** of the second kind and poly-Cauchy **polynomials** of the second kind. From the properties of Sheﬀer sequences of these **polynomials** arising from umbral calculus, we derive new and interesting identities.

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In this paper, by using p-adic Volkenborn integral, and generating functions, we give some properties of the Bernstein basis functions, the Apostol-**Daehee** numbers and **polynomials**, Apostol-Bernoulli **polynomials**, some special numbers including the Stirling numbers, the Euler numbers, the **Daehee** numbers, and the Changhee numbers. By using an integral equation and functional equations of the generating functions and their partial diﬀerential equations (PDEs), we give a recurrence relation for the Apostol-**Daehee** **polynomials**. We also give some identities, relations, and integral representations for these numbers and **polynomials**. By using these relations, we compute these numbers and **polynomials**. We make further remarks and

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The p-adic q-integral (or q-Volkenborn integration) was deﬁned by Kim (see [, ]). From p-adic q-integral equations, we can derive various q-extensions of Bernoulli polyno- mials and numbers (see [–]). In [], DS Kim and T Kim studied **Daehee** **polynomials** and numbers and their applications. In [], Kim et al. introduced the q-analogue of **Daehee** numbers and **polynomials** which are called q-**Daehee** numbers and **polynomials**. Lim con- sidered in [] the modiﬁed q-**Daehee** numbers and **polynomials** which are diﬀerent from the q-**Daehee** numbers and **polynomials** of Kim et al. In this paper, we consider (h, q)- **Daehee** numbers and **polynomials** and give some interesting identities. In case h = , we cover the q-analogue of **Daehee** numbers and **polynomials** of Kim et al. (see []). In case h = , we have modiﬁed q-**Daehee** numbers and **polynomials** in []. We can ﬁnd out vari- ous (h, q)-related numbers and **polynomials** in [, , ].

We established a new operator identity for the general case of the Jacobi **polynomials** which is a kind of operator disentanglement and insofar it is related to reordering of non-commuting operators to normal ordering (all differential operators behind the multiplication operators) and is important and well known in quantum optics for the annihilation and creation operators of the Heisenberg- Weyl group and also in the theory of differential equations. Operator identities can be applied to arbitrary functions and they provide then function identities. In this way we could prove a kind of convolution theorem for the Jacobi poly- nomials with a certain similarity to the Vandermond convolution identity for binomial coefficients. Sometimes it was difficult to find out within the immense literature to **polynomials** whether or not a particular formula or approach is already known or is it novel and our main attention was directed to the correct- ness of the formulae.

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In this paper, we have studied the harmonic, the hyperharmonic, the **Daehee** and the higher-order **Daehee** numbers which are diﬀerent from the previous research articles. In Sect. 2, we present some elementary identities between the harmonic and the hyperhar- monic numbers. In Sect. 3, we study some relations and properties for the harmonic and the hyperharmonic numbers, the **Daehee** and the higher-order **Daehee** numbers. Addi- tionally, the derangement numbers and the Cauchy numbers are also studied in Sect. 3. In Sect. 4, we study a nonlinear diﬀerential equation arising from the generating function of the harmonic numbers and we give some identities of harmonic and hyperharmonic num- bers, the **Daehee** and higher-order **Daehee** numbers which are derived from this nonlinear diﬀerential equation.

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Abstract. In this paper, we introduce a new class of (p, q)-analogue type of Fubini numbers and **polynomials** and investigate some properties of these **polynomials**. We establish summation formulas of these **polynomials** by summation techniques series. Furthermore, we consider some relationships for (p, q)-Fubini **polynomials** associated with (p, q)-Bernoulli **polynomials**, (p, q)-Euler **polynomials** and (p, q)-Genocchi poly- nomials and (p, q)-Stirling numbers of the second kind.

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The purpose of this paper is to introduce and investigate new unification of unified family of Apostol-type **polynomials** and numbers based on results given in [1] [2]. Also, we derive some properties for these **polynomials** and obtain some relationships between the Jacobi **polynomials**, Laguerre **polynomials**, Hermite **polynomials**, Stirling numbers and some other types of genera- lized **polynomials**.

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The Hecke operators have many applications in various spaces like the space of elliptic modular forms, the space of **polynomials** and others. Many mathematicians applied them to obtain applications in analytic number theory, harmonic analysis, theoretical physics, equidistribution of Hecke points on a family of homogeneous varieties, and cohomology. For instance, Hecke operators are used to investigate and study Fourier coeﬃcients of modular forms, to explore other properties of the Hecke-eigenforms, which satisfy many interesting arithmetic relations. For more details on Hecke operators, see [, ]. Recently, the Hurwitz zeta functions and the Apostol-Bernoulli **polynomials** have been studied by many authors, for example, see (cf. [–], the others).

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When it comes to Genocchi numbers, the most common thing comes to our mind is to research the relations between Genocchi numbers, Bernoulli numbers [14–16] and Euler numbers [14, 17]. Indeed, most researches on Genocchi numbers concern the relations between these three kinds of numbers (see for example [2–4, 18, 19]). In other words, there are many literatures that provide identities on these three kinds of numbers. Similarly, when it comes to Genocchi **polynomials**, the most common thing is to research on the relations between Genocchi **polynomials**, Bernoulli **polynomials** and Euler **polynomials** (see for example [2–4, 9, 18–21]). Even though when it comes to the generalized Genocchi numbers and generalized Genocchi **polynomials**, it is unavoidable to research the relations as above.

2. Berndt, BC: Periodic Bernoulli numbers, summation formulas and applications. In: Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975). Publication of the Mathematics Research Center, the University of Wisconsin, vol. 35, pp. 143-189. Academic Press, New York (1975) 3. Carlitz, L: A note on Bernoulli numbers and **polynomials**. Elem. Math. 29, 90-92 (1974)

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Many mathematical entities and objects are attributed to (and named after) him. These entities and objects include (among other items) Srivastava’s **polynomials** and func- tions, Carlitz-Srivastava **polynomials**, Srivastava-Buschman **polynomials**, Srivastava- Singhal **polynomials**, Chan-Chyan-Srivastava **polynomials**, Erkuş-Srivastava **polynomials**, Srivastava-Daoust multivariable hypergeometric function, Srivastava-Panda multivari- able H-function, Singhal-Srivastava generating function, Srivastava-Agarwal basic (or q-) generating function, and Wu-Srivastava inequality in the ﬁeld of higher transcendental functions; Srivastava-Owa, Choi-Saigo-Srivastava, Jung-Kim-Srivastava, Liu-Srivastava, Cho-Kwon-Srivastava, Dziok-Srivastava, Srivastava-Attiya and Srivastava-Wright oper- ators in the ﬁeld of geometric function theory in complex analysis; Srivastava-Gupta operator in the ﬁeld of approximation theory; the Srivastava, Adamchik-Srivastava and Choi-Srivastava methods in the ﬁeld of analytic number theory; and so on.

(1.9) If we take a = 1, b = c = e in (1.7), (1.8) and (1.9) respectively, we have (1.4), (1.5) and (1.6). Obviously, when we set λ = 1, α = 1, a = 1, b = c = e in (1.7), (1.8) and (1.9), we have classical Bernoulli **polynomials** B n (x), classical Euler

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We consider the Witt-type formula for the nth twisted **Daehee** numbers and **polynomials** and investigate some properties of those numbers and **polynomials**. In particular, the nth twisted **Daehee** numbers are closely related to higher-order Bernoulli numbers and Bernoulli numbers of the second kind.

In this paper, we consider the Barnes-type **Daehee** with λ -parameter and degenerate Euler mixed-type **polynomials**. We present several explicit formulas and recurrence relations for these **polynomials**. Also, we establish a connection between our **polynomials** and several known families of **polynomials**.

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