A q-analogue of r-Whitney numbers of the second kind, denoted by W_m,r[n,k]_q, is defined by mean... more A q-analogue of r-Whitney numbers of the second kind, denoted by W_m,r[n,k]_q, is defined by means of a triangular recurrence relation. In this paper, several fundamental properties for the q-analogue are established including other forms of recurrence relations, explicit formulas and generating functions. Moreover, a kind of Hankel transform for W_m,r[n,k]_q is obtained.
In this paper, more identities for generalized poly-Euler and poly-Bernoulli polynomials with thr... more In this paper, more identities for generalized poly-Euler and poly-Bernoulli polynomials with three parameters are obtained in connection with the Stirling numbers of the second kind. Moreover, symmetrized generalization is introduced to establish certain duality relation.
This paper presents a formula for the distinct dissections by diagonals of a regular n-gon modulo... more This paper presents a formula for the distinct dissections by diagonals of a regular n-gon modulo the action of the dihedral group. This counting includes dissection with intersecting or non-intersecting diagonals. We utilize a corollary of the CauchyFrobenius theorem, which involves counting of cycles. We also give an explicit formula for the prime number case. We give as a remark the number of distinct dissections, modulo the action of the cyclic group of finite order.
In this paper, a q-analogue of r-Whitney-Lah numbers, also known as (q,r)-Whitney-Lah number, den... more In this paper, a q-analogue of r-Whitney-Lah numbers, also known as (q,r)-Whitney-Lah number, denoted by $L_{m,r} [n, k]_q$ is defined using the triangular recurrence relation. Several fundamental properties for the q-analogue are established such as vertical and horizontal recurrence relations, horizontal and exponential generating functions. Moreover, an explicit formula for (q, r)-Whitney-Lah number is derived using the concept of q-difference operator, particularly, the q-analogue of Newton’s Interpolation Formula (the umbral version of Taylor series). Furthermore, an explicit formula for the first form (q, r)-Dowling numbers is obtained which is expressed in terms of (q,r)-Whitney-Lah numbers and (q,r)-Whitney numbers of the second kind.
Fractional and fractal derivatives are both generalizations of the usual derivatives that conside... more Fractional and fractal derivatives are both generalizations of the usual derivatives that consider derivatives of non-integer orders. Interest in these generalizations has been triggered by a resurgence of clamor to develop a mathematical tool to describe “roughness” in the spirit of Mandelbrot’s (1967) fractal geometry. Fractional derivatives take the analytic approach towards developing a rational order derivative while fractal derivatives follow a more concrete, albeit geometric approach to the same end. Since both approaches alleged to extend whole derivatives to rational derivatives, it is not surprising that confusion will arise over which generalization to use in practice. This paper attempts to highlight the connection between the various generalizations to fractional and fractal derivatives with the end-in-view of making these concepts useful in various physics applications and to resolve some of the confusion that arise out of the fundamental philosophical differences in the derivation of fractional derivatives (non-local concept) and fractal derivatives (local concept).
A q-analogue of r-Whitney numbers of the second kind, denoted by W m,r [n, k] q , is defined by m... more A q-analogue of r-Whitney numbers of the second kind, denoted by W m,r [n, k] q , is defined by means of a triangular recurrence relation. In this paper, several fundamental properties for the said q-analogue are established including other forms of recurrence relations, explicit formulas and generating functions. Moreover, a kind of Hankel transform for W m,r [n, k] q is obtained.
In this paper, using the rational generating for the second form of the q-analogue of r-Whitney n... more In this paper, using the rational generating for the second form of the q-analogue of r-Whitney numbers of the second kind, certain divisibility property for this form is established. Moreover, the Hankel transform for the second form of the q-analogue of r-Dowling numbers is derived.
This paper derives another form of explicit formula for $(r,\beta)$-Bell numbers using the Faa di... more This paper derives another form of explicit formula for $(r,\beta)$-Bell numbers using the Faa di Bruno's formula and certain identity of Bell polynomials of the second kind. This formula is expressed in terms of the $r$-Whitney numbers of the second kind and the ordinary Lah numbers. As a consequence, a relation between $(r,\beta)$-Bell numbers and the sums of row entries of the product of two matrices containing the $r$-Whitney numbers of the second kind and the ordinary Lah numbers is established. Moreover, a $q$-analogue of the explicit formula is obtained.
A subset D of a group G is called a D-set if every element of G which is not in D has its inverse... more A subset D of a group G is called a D-set if every element of G which is not in D has its inverse in D. In this paper, we gave some of the properties of a D-set.
International Journal of Mathematics and Mathematical Sciences, 2015
We define two forms ofq-analogue of noncentral Stirling numbers of the second kind and obtain som... more We define two forms ofq-analogue of noncentral Stirling numbers of the second kind and obtain some properties parallel to those of noncentral Stirling numbers. Certain combinatorial interpretation is given for the second form of theq-analogue in the context of 0-1 tableaux which, consequently, yields certain additive identity and some convolution-type formulas. Finally, aq-analogue of noncentral Bell numbers is defined and its Hankel transform is established.
Asymptotic approximations of Tangent polynomials, Tangent-Bernoulli, and Tangent-Genocchi polynom... more Asymptotic approximations of Tangent polynomials, Tangent-Bernoulli, and Tangent-Genocchi polynomials are derived using saddle point method and the approximations are expressed in terms of hyperbolic functions. For each polynomial there are two approximations derived with one having enlarged region of validity.
A q-analogue of r-Whitney numbers of the second kind, denoted by W_m,r[n,k]_q, is defined by mean... more A q-analogue of r-Whitney numbers of the second kind, denoted by W_m,r[n,k]_q, is defined by means of a triangular recurrence relation. In this paper, several fundamental properties for the q-analogue are established including other forms of recurrence relations, explicit formulas and generating functions. Moreover, a kind of Hankel transform for W_m,r[n,k]_q is obtained.
In this paper, more identities for generalized poly-Euler and poly-Bernoulli polynomials with thr... more In this paper, more identities for generalized poly-Euler and poly-Bernoulli polynomials with three parameters are obtained in connection with the Stirling numbers of the second kind. Moreover, symmetrized generalization is introduced to establish certain duality relation.
This paper presents a formula for the distinct dissections by diagonals of a regular n-gon modulo... more This paper presents a formula for the distinct dissections by diagonals of a regular n-gon modulo the action of the dihedral group. This counting includes dissection with intersecting or non-intersecting diagonals. We utilize a corollary of the CauchyFrobenius theorem, which involves counting of cycles. We also give an explicit formula for the prime number case. We give as a remark the number of distinct dissections, modulo the action of the cyclic group of finite order.
In this paper, a q-analogue of r-Whitney-Lah numbers, also known as (q,r)-Whitney-Lah number, den... more In this paper, a q-analogue of r-Whitney-Lah numbers, also known as (q,r)-Whitney-Lah number, denoted by $L_{m,r} [n, k]_q$ is defined using the triangular recurrence relation. Several fundamental properties for the q-analogue are established such as vertical and horizontal recurrence relations, horizontal and exponential generating functions. Moreover, an explicit formula for (q, r)-Whitney-Lah number is derived using the concept of q-difference operator, particularly, the q-analogue of Newton’s Interpolation Formula (the umbral version of Taylor series). Furthermore, an explicit formula for the first form (q, r)-Dowling numbers is obtained which is expressed in terms of (q,r)-Whitney-Lah numbers and (q,r)-Whitney numbers of the second kind.
Fractional and fractal derivatives are both generalizations of the usual derivatives that conside... more Fractional and fractal derivatives are both generalizations of the usual derivatives that consider derivatives of non-integer orders. Interest in these generalizations has been triggered by a resurgence of clamor to develop a mathematical tool to describe “roughness” in the spirit of Mandelbrot’s (1967) fractal geometry. Fractional derivatives take the analytic approach towards developing a rational order derivative while fractal derivatives follow a more concrete, albeit geometric approach to the same end. Since both approaches alleged to extend whole derivatives to rational derivatives, it is not surprising that confusion will arise over which generalization to use in practice. This paper attempts to highlight the connection between the various generalizations to fractional and fractal derivatives with the end-in-view of making these concepts useful in various physics applications and to resolve some of the confusion that arise out of the fundamental philosophical differences in the derivation of fractional derivatives (non-local concept) and fractal derivatives (local concept).
A q-analogue of r-Whitney numbers of the second kind, denoted by W m,r [n, k] q , is defined by m... more A q-analogue of r-Whitney numbers of the second kind, denoted by W m,r [n, k] q , is defined by means of a triangular recurrence relation. In this paper, several fundamental properties for the said q-analogue are established including other forms of recurrence relations, explicit formulas and generating functions. Moreover, a kind of Hankel transform for W m,r [n, k] q is obtained.
In this paper, using the rational generating for the second form of the q-analogue of r-Whitney n... more In this paper, using the rational generating for the second form of the q-analogue of r-Whitney numbers of the second kind, certain divisibility property for this form is established. Moreover, the Hankel transform for the second form of the q-analogue of r-Dowling numbers is derived.
This paper derives another form of explicit formula for $(r,\beta)$-Bell numbers using the Faa di... more This paper derives another form of explicit formula for $(r,\beta)$-Bell numbers using the Faa di Bruno's formula and certain identity of Bell polynomials of the second kind. This formula is expressed in terms of the $r$-Whitney numbers of the second kind and the ordinary Lah numbers. As a consequence, a relation between $(r,\beta)$-Bell numbers and the sums of row entries of the product of two matrices containing the $r$-Whitney numbers of the second kind and the ordinary Lah numbers is established. Moreover, a $q$-analogue of the explicit formula is obtained.
A subset D of a group G is called a D-set if every element of G which is not in D has its inverse... more A subset D of a group G is called a D-set if every element of G which is not in D has its inverse in D. In this paper, we gave some of the properties of a D-set.
International Journal of Mathematics and Mathematical Sciences, 2015
We define two forms ofq-analogue of noncentral Stirling numbers of the second kind and obtain som... more We define two forms ofq-analogue of noncentral Stirling numbers of the second kind and obtain some properties parallel to those of noncentral Stirling numbers. Certain combinatorial interpretation is given for the second form of theq-analogue in the context of 0-1 tableaux which, consequently, yields certain additive identity and some convolution-type formulas. Finally, aq-analogue of noncentral Bell numbers is defined and its Hankel transform is established.
Asymptotic approximations of Tangent polynomials, Tangent-Bernoulli, and Tangent-Genocchi polynom... more Asymptotic approximations of Tangent polynomials, Tangent-Bernoulli, and Tangent-Genocchi polynomials are derived using saddle point method and the approximations are expressed in terms of hyperbolic functions. For each polynomial there are two approximations derived with one having enlarged region of validity.
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