A new class of refined eulerian polynomials

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# A new class of refined eulerian polynomials

In combinatorics, the Eulerian Number A n, mis the number of permutations of the numbers 1 to n in which exactly m elements are greater than previous element. For example, there are 4 permutations of the number 1 to 3 in which exactly 1 element is greater than the previous elements. The Eulerian polynomials are defined by the exponential generating function. The Eulerian polynomials can be computed by the recurrence. An explicit formula for A n, m is. We can calculate A n, m by recurrence relation:.

Attention reader! If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute. See your article appearing on the GeeksforGeeks main page and help other Geeks. Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below. Writing code in comment? Please use ide. Python3 Program to find Eulerian number A n, m. Return euleriannumber A n, m. WriteLine eulerian n, m. Python3 Program to find Eulerian.

For each row from 1 to n. For each column from 0 to m. If i is greater than j. If j is 0, then make that. This code is contributed by Prasad Kshirsagar. Check out this Author's contributed articles.

Load Comments. We use cookies to ensure you have the best browsing experience on our website.Some implicit summation formulae and general symmetry identities are derived using different analytical means and applying generating functions. This is a preview of subscription content, log in to check access. Rent this article via DeepDyve. Bell E. Betti G. Dattoli G. Rendiconti di Mathematica 19— Dere R. Russian J.

Khan S. A, Hassan N. Kurt B. Luo Q-M. Luo Q. Milne Thomsons L. Pathan M. Pathan, M. East—West J. Qui F. Google Scholar. Simsek Y:: Generating functions for generalized Stirling type numbers, array type polynomials, Eulerian type polynomials and their applications.

Fixed Point Theory Appl. Simsek Y. Srivastava H. Ellis Horwood Limited. Tuenter H. Yang S. Zhang Z. Download references. Correspondence to M.Tremblay, S. Gaboury, B. The main object of this paper is to introduce and investigate two new classes of generalized Apostol-Euler and Apostol-Genocchi polynomials. In particular, we obtain a new addition formula for the new class of the generalized Apostol-Euler polynomials. The literature contains a large number of interesting properties and relationships involving these polynomials [ 14 — 7 ].

These appear in many applications in combinatorics, number theory, and numerical analysis. Many interesting extensions to these polynomials have been given. These polynomials are defined, respectively, as follows. Definition 1. The reader should read [ 11 — 20 ]. Moreover, Srivastava et al. They investigated the following forms. Lately, Kurt [ 23 ] presented a new interesting class of generalized Euler polynomials.

Explicitly, he introduced the next definition. For the new class of Apostol-Euler polynomials, we establish a new addition theorem with the help of a result given by Srivastava et al.

Finally, we exhibit some relationships between the generalized Apostol-Euler polynomials and other polynomials or special functions with the help of the new addition formula. Definition 2. These are stated as Theorems 2. Theorem 2. Considering the generating function 2. Proof of 2. Remark 2. In this section, we establish a new addition theorem for the generalized Apostol-Euler polynomials.

## A New Class of Generalized Polynomials Associated with Hermite and Euler Polynomials

This new formula is based on a result due to Srivastava et al. The next theorem has been invented by Srivastava et al. However, the theorem is given without proof see [ 24pages —]. Theorem 3. Remark 3. The choice of 1 in the conditions of 3. Now, applying the last theorem with special choices of functions and parameters furnishes the next very interesting addition formula.

This formula is contained in the following corollary. Corollary 3. We end this section by giving two interesting relationships involving the new addition formula 3. ### Donate to arXiv

Some features of the site may not work correctly. DOI: Sun and Y. Wang and H. SunY. WangH. View on Springer. Save to Library. Create Alert. Launch Research Feed. Share This Paper. Cai, Qing-Hu Hou, … A. Yang Sun Yang Discret. Citation Type. Has PDF. Publication Type. More Filters. View 1 excerpt, cites background. Research Feed. Antichain generating polynomials of posets. Around the q-binomial-Eulerian polynomials. Some spectral invariants of the neighborhood corona of graphs. On two unimodal descent polynomials. References Publications referenced by this paper. On the Unimodality of some Partition Polynomials.Abstract: The Ehrhart polynomial of a lattice polytope encodes information about the number of integer lattice points in positive integral dilates of.

The -polynomial of is the numerator polynomial of the generating function of its Ehrhart polynomial. A zonotope is any projection of a higher dimensional cube. We give a combinatorial description of the -polynomial of a lattice zonotope in terms of refined descent statistics of permutations and prove that the -polynomial of every lattice zonotope has only real roots and therefore unimodal coefficients.

Furthermore, we present a closed formula for the -polynomial of a zonotope in matroidal terms which is analogous to a result by Stanley on the Ehrhart polynomial.

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Our results hold not only for -polynomials but carry over to general combinatorial positive valuations. Moreover, we give a complete description of the convex hull of all -polynomials of zonotopes in a given dimension: it is a simplicial cone spanned by refined Eulerian polynomials. References [Enhancements On Off] What's this? The third author was partially supported by the U.

SamCorrigendum to: Positivity theorems for solid-angle polynomials [ ]Beitr. Algebra Geom. Betke and P. GritzmannAn application of valuation theory to two problems in discrete geometryDiscrete Math. EhrenborgM. Readdyand E. Theory Ser. A 81no. Paris— French. MR  Takayuki HibiDual polytopes of rational convex polytopesCombinatorica 12no.

JEMS 20no. ZieglerBounds for lattice polytopes containing a fixed number of interior points in a sublatticeCanad. McMullenValuations and Euler-type relations on certain classes of convex polytopesProc. London Math. MR  Thomas W.The Eulerian polynomials are defined by the exponential generating function. The Eulerian polynomials can be computed by the recurrence. An equivalent way to write this definition is to set the Eulerian polynomials inductively by.

The definition given is used by major authors like D.

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Knuth, D. Foata and F. In the older literature for example in L. Comtet, Advanced Combinatorics a slightly different definition is used, namely. Frobenius proved that the Eulerian polynomials are equal to:. Euler's definition A n, k is A Using the Iverson bracket notation [. The combinatorial interpretation of the Eulerian polynomials is then given by.

Leonhard Euler introduced the polynomials in  in the form. Euler introduced the Eulerian polynomials in an attempt to evaluate the Dirichlet eta function. This led him to conjecture the functional equation of the eta function which immediately implies the functional equation of the zeta function. Most simply put, the relation Euler was after was. Though Euler's reasoning was not rigorous by modern standards it was a milestone on the way to Riemann's proof of the functional equation of the zeta function.

The facsimile shows Eulerian polynomials as given by Euler in his work Institutiones calculi differentialis It is interesting to note that the original definition of Euler coincides with the definition in the DLMF. Many elementary classes of sequences have an Eulerian generating function.

Matthias Beck - Classification of combinatorial polynomials

A few examples are collocated in the table below. Eulerian polynomials A n x and Euler polynomials E n x have a sequence of values in common up to a binary shift. The formulas below show how rich in content the Eulerian polynomials are. For nonpositive integer values of s, the polylogarithm is a rational function. The first few are.

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Serkan Araci.

## Polynomials with palindromic and unimodal coefficients

A class of generating functions for a new generalization of Eulerian polynomials with their interpolation function. We derive useful results involving these Eulerian polynomials including for example their generating functions, new series and L-type functions. Eulerian polynomials, Fermionic p-adic q-integral on ZpMellin transformation, L-functions. Recently, Kim et al have studied on some identities of the Eulerian polynomials in connec- tion with Genocchi and Tangent numbers using the fermionic p-adic integral on Zp in .

Kim and Kim introduced a new de…nition of Eulerian polynomials and gave their symmetric relations for details, see , . Araci et al also introduced the generalizations of the Eulerian-type polynomials using the fermionic p-adic q-integral on Zp and derived some new interesting identities cf.

Hereby, we note that generating functions transform problems about sequences into problems about polynomials. By this way, generating functions are important to solve all sorts of counting problems. Let p be a …xed odd prime number. Throughout this paper, we always make use of the following notations: Zp denotes the ring of p-adic rational integers, Q denotes the …eld of rational numbers, Qp denotes the …eld of p-adic rational numbers, and Cp denotes the completion of algebraic closure of Qp.

We note that limq! It follows from the Eq. Kim and M. Actually, we are motivated from the papers of Kim et al  and Kim et al  to write this paper. Then we consider the following equality by using the Eq. Theorem 1. Theorem 2. Theorem 3. Theorem 4. The following identity holds true: 1 n An; q; w 2. The values of the negative integer points, also found by Euler, are rational numbers and play a vital and important role in the theory of modular forms. Many generalization of the Riemann zeta function, such as Dirichlet series, Dirichlet L-functions and L-functions, are worked in , , , , , , .

In this …nal part, our objective is to introduce a new generalization of the Eulerian-L func- tion applying Mellin transformation to the generating function of the Eulerian polynomials.

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From the Eq. Theorem 5. References  S. 