This paper presents the evaluation of the Euler sums of generalized hyperharmonic numbers H(p,q)n ?H(p,q)(r) = ?Xn=1 H(p,q)n/nr in terms of the famous Euler sums of generalized harmonic numbers. Moreover, several infinite series, whose terms consist of certain harmonic numbers and reciprocal binomial coefficients, are evaluated in terms of the Riemann zeta values.
{"title":"Euler sums of generalized harmonic numbers and connected extensions","authors":"M. Can, L. Kargin, A. Dil, G. Soylu","doi":"10.2298/aadm210122014c","DOIUrl":"https://doi.org/10.2298/aadm210122014c","url":null,"abstract":"This paper presents the evaluation of the Euler sums of generalized hyperharmonic numbers H(p,q)n ?H(p,q)(r) = ?Xn=1 H(p,q)n/nr in terms of the famous Euler sums of generalized harmonic numbers. Moreover, several infinite series, whose terms consist of certain harmonic numbers and reciprocal binomial coefficients, are evaluated in terms of the Riemann zeta values.","PeriodicalId":51232,"journal":{"name":"Applicable Analysis and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68353667","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the main umbral operators J, M and N associated with the Cayley continuants U(?)n(x) and the generalized Sylvester continuants H(?)n(x) = U(?+n)n(x). In particular, we obtain their representation in terms of the differential operator Dx and the shift operator E. Then, by using these representations, we obtain some combinatorial and differential identities for the continuants U(?)n(x) and H(?)n(x).
研究了Cayley连续体U(?) N (x)和广义Sylvester连续体H(?) N (x) = U(?+ N) N (x)的主本影算子J、M和N。特别地，我们得到了它们的微分算子Dx和移位算子e的表示，然后利用这些表示，我们得到了连续体U(?)n(x)和H(?)n(x)的一些组合恒等式和微分恒等式。
{"title":"Umbral operators for Cayley and Sylvester continuants","authors":"E. Munarini","doi":"10.2298/aadm200120037m","DOIUrl":"https://doi.org/10.2298/aadm200120037m","url":null,"abstract":"We study the main umbral operators J, M and N associated with the Cayley continuants U(?)n(x) and the generalized Sylvester continuants H(?)n(x) = U(?+n)n(x). In particular, we obtain their representation in terms of the differential operator Dx and the shift operator E. Then, by using these representations, we obtain some combinatorial and differential identities for the continuants U(?)n(x) and H(?)n(x).","PeriodicalId":51232,"journal":{"name":"Applicable Analysis and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68353308","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider various statistics on the set Fn consisting of the distinct permutations of length n+1 that arise as a flattening of some partition of the same size. In particular, we enumerate members of Fn according to the number of occurrences of three-letter consecutive patterns, considered more broadly in the context of r-partitions. As special cases of our results, we obtain formulas for the number of members of Fn avoiding a given consecutive pattern and for the total number of occurrences of a pattern over all members of Fn.
{"title":"Counting subword patterns in permutations arising as flattened partitions of sets","authors":"T. Mansour, M. Shattuck","doi":"10.2298/aadm210223009m","DOIUrl":"https://doi.org/10.2298/aadm210223009m","url":null,"abstract":"We consider various statistics on the set Fn consisting of the distinct permutations of length n+1 that arise as a flattening of some partition of the same size. In particular, we enumerate members of Fn according to the number of occurrences of three-letter consecutive patterns, considered more broadly in the context of r-partitions. As special cases of our results, we obtain formulas for the number of members of Fn avoiding a given consecutive pattern and for the total number of occurrences of a pattern over all members of Fn.","PeriodicalId":51232,"journal":{"name":"Applicable Analysis and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68353680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study a boundary value problem for the q-Dirac equation and eigenvalue-dependent boundary conditions. We introduce a self-adjoint operator in a suitable Hilbert space and illustrate the boundary value problem as a spectral problem for this operator. We investigate the properties of the eigenvalues and vector-valued eigenfunctions. We construct Green?s function.
{"title":"A q-Dirac boundary value problem with eigenparameter-dependent boundary conditions","authors":"M. Bohner, Ayça Çetinkaya","doi":"10.2298/aadm220323036b","DOIUrl":"https://doi.org/10.2298/aadm220323036b","url":null,"abstract":"We study a boundary value problem for the q-Dirac equation and eigenvalue-dependent boundary conditions. We introduce a self-adjoint operator in a suitable Hilbert space and illustrate the boundary value problem as a spectral problem for this operator. We investigate the properties of the eigenvalues and vector-valued eigenfunctions. We construct Green?s function.","PeriodicalId":51232,"journal":{"name":"Applicable Analysis and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68354195","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Making use of complex analytic techniques as well as methods involving weight functions, we study a few equivalent conditions of a Hilbert-type integral inequality with nonhomogeneous kernel and parameters. In the form of applications we deduce a few equivalent conditions of a Hilbert-type integral inequality with homogeneous kernel, and we additionally consider operator expressions.
{"title":"An equivalent property of a Hilbert-type integral inequality and its applications","authors":"B. Yang, D. Andrica, O. Bagdasar, M. Rassias","doi":"10.2298/aadm220514025y","DOIUrl":"https://doi.org/10.2298/aadm220514025y","url":null,"abstract":"Making use of complex analytic techniques as well as methods involving weight functions, we study a few equivalent conditions of a Hilbert-type integral inequality with nonhomogeneous kernel and parameters. In the form of applications we deduce a few equivalent conditions of a Hilbert-type integral inequality with homogeneous kernel, and we additionally consider operator expressions.","PeriodicalId":51232,"journal":{"name":"Applicable Analysis and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68354284","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we provide a systematic way to study on some general Wilker-Huygens type inequalities for the trigonometric and hyperbolic functions, lemniscate and hyperbolic lemniscate functions, and their corresponding inverse functions. Our results are some extensions and refinements of the recently published results in [A. Mhanna, On a general Huygens-Wilker inequality, Appl. Math. E.-Notes, 20 (2020), 79-81; MR4076436], and improve many previous results involving Wilker-Huygens type inequalities.
{"title":"Some general Wilker-Huygens inequalities","authors":"Tie-hong Zhao, Yu‐ming Chu","doi":"10.2298/aadm210518032z","DOIUrl":"https://doi.org/10.2298/aadm210518032z","url":null,"abstract":"In this paper, we provide a systematic way to study on some general Wilker-Huygens type inequalities for the trigonometric and hyperbolic functions, lemniscate and hyperbolic lemniscate functions, and their corresponding inverse functions. Our results are some extensions and refinements of the recently published results in [A. Mhanna, On a general Huygens-Wilker inequality, Appl. Math. E.-Notes, 20 (2020), 79-81; MR4076436], and improve many previous results involving Wilker-Huygens type inequalities.","PeriodicalId":51232,"journal":{"name":"Applicable Analysis and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68353821","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A. Acu, M. Dancs, M. Heilmann, Vlad Paşca, I. Raşa
We consider a sequence of positive linear operators Ln of Bernstein-Schnabl type. It was studied in the literature from various points of view; we provide new properties of it. The eigenstructure of these operators is described. We investigate the kernel of Ln which is related with the set of solutions of a difference equation. Several algorithms are proposed in order to solve the involved problems.
{"title":"A Bernstein-Schnabl type operator: Applications to difference equations","authors":"A. Acu, M. Dancs, M. Heilmann, Vlad Paşca, I. Raşa","doi":"10.2298/aadm210714011a","DOIUrl":"https://doi.org/10.2298/aadm210714011a","url":null,"abstract":"We consider a sequence of positive linear operators Ln of Bernstein-Schnabl type. It was studied in the literature from various points of view; we provide new properties of it. The eigenstructure of these operators is described. We investigate the kernel of Ln which is related with the set of solutions of a difference equation. Several algorithms are proposed in order to solve the involved problems.","PeriodicalId":51232,"journal":{"name":"Applicable Analysis and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68353905","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let P(z):= ?nv=0 avzv be a univariate complex coefficient polynomial of degree n. It was shown by Malik [J London Math Soc, 1 (1969), 57-60] that if P(z) has all its zeros in |z| ? k, k ? 1, then max|z|=1 |P?(z)| ? n 1 + k max |z|=1 |P(z)|. In this paper, we prove an inequality for the polar derivative of a polynomial which besides give extensions and refinements of the above inequality also produce various inequalities that are sharper than the previous ones known in very rich literature on this subject.
设P(z):= ?nv=0 avzv是n次的单变量复系数多项式。Malik [J London Math Soc, 1(1969)， 57-60]证明了如果P(z)的所有零都在|z| ?K, K ?1，则max|z|=1 |P?(z)| ?n 1 + k max |z|=1 |P(z)|。在本文中，我们证明了一个多项式的极坐标导数的不等式，该不等式除了给出上述不等式的推广和改进外，还产生了各种不等式，这些不等式比以前在非常丰富的文献中已知的不等式更尖锐。
{"title":"Note on an inequality of M.A. Malik","authors":"A. Mir, Abrar Ahmad, A. Malik","doi":"10.2298/aadm210529030m","DOIUrl":"https://doi.org/10.2298/aadm210529030m","url":null,"abstract":"Let P(z):= ?nv=0 avzv be a univariate complex coefficient polynomial of degree n. It was shown by Malik [J London Math Soc, 1 (1969), 57-60] that if P(z) has all its zeros in |z| ? k, k ? 1, then max|z|=1 |P?(z)| ? n 1 + k max |z|=1 |P(z)|. In this paper, we prove an inequality for the polar derivative of a polynomial which besides give extensions and refinements of the above inequality also produce various inequalities that are sharper than the previous ones known in very rich literature on this subject.","PeriodicalId":51232,"journal":{"name":"Applicable Analysis and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68353843","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The aim of this paper is to provide new refinements of Huygens-Wilker-Lazarovic inequalities using hyperbolic cosine polynomials. We give an unitary approach for both inequalities of trigonometric and hyperbolic functions.
{"title":"Refinements of Huygens-Wilker-Lazarovic inequalities via the hyperbolic cosine polynomials","authors":"G. Bercu","doi":"10.2298/aadm200403004b","DOIUrl":"https://doi.org/10.2298/aadm200403004b","url":null,"abstract":"The aim of this paper is to provide new refinements of Huygens-Wilker-Lazarovic inequalities using hyperbolic cosine polynomials. We give an unitary approach for both inequalities of trigonometric and hyperbolic functions.","PeriodicalId":51232,"journal":{"name":"Applicable Analysis and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68353494","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we consider the function fp(t) = ? 2p?2(?2pt + p;p), where ?2(x;n) defined by ?2(x;p) = 2?p/2/?(p/2) e?x/2xp/2?1, is the density function of a ?2-distribution with n degrees of freedom. The asymptotic expansion of fp(t) for p ? ?, where p is not necessarily an integer, is obtained by an application of the standard asymptotics of ln ?(x). Two different methods of obtaining the coefficients in the asymptotic expansion are presented, which involve the use of the Bell polynomials.
本文考虑函数fp(t) = ?2 p 2 (?2 pt + p, p)在哪里? 2 (x; n)定义的? 2 (x, p) = 2 ? p / 2 / ? ? (p / 2) e x / 2 xp / 2 ?1，是n个自由度的?2分布的密度函数。p(t)的渐近展开式其中p不一定是整数，它是通过应用ln ?(x)的标准渐近得到的。给出了两种不同的求渐近展开系数的方法，这两种方法都涉及到贝尔多项式的使用。
{"title":"Complete asymptotic expansions related to the probability density function of the χ2-distribution","authors":"Chao Chen, H. Srivastava","doi":"10.2298/aadm210720015c","DOIUrl":"https://doi.org/10.2298/aadm210720015c","url":null,"abstract":"In this paper, we consider the function fp(t) = ? 2p?2(?2pt + p;p), where ?2(x;n) defined by ?2(x;p) = 2?p/2/?(p/2) e?x/2xp/2?1, is the density function of a ?2-distribution with n degrees of freedom. The asymptotic expansion of fp(t) for p ? ?, where p is not necessarily an integer, is obtained by an application of the standard asymptotics of ln ?(x). Two different methods of obtaining the coefficients in the asymptotic expansion are presented, which involve the use of the Bell polynomials.","PeriodicalId":51232,"journal":{"name":"Applicable Analysis and Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68353943","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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