scholarly journals A Unified Generalization of the Catalan, Fuss, and Fuss—Catalan Numbers

2019 ◽  
Vol 24 (2) ◽  
pp. 49
Author(s):  
Feng Qi ◽  
Xiao-Ting Shi ◽  
Pietro Cerone
Keyword(s):  
Second Order ◽  
Catalan Numbers ◽  
Integral Representations ◽  
Complete Monotonicity ◽  
Product Ratio ◽  
Special Case

In the paper, the authors introduce a unified generalization of the Catalan numbers, the Fuss numbers, the Fuss–Catalan numbers, and the Catalan–Qi function, and discover some properties of the unified generalization, including a product-ratio expression of the unified generalization in terms of the Catalan–Qi functions, three integral representations of the unified generalization, and the logarithmically complete monotonicity of the second order for a special case of the unified generalization.

scholarly journals An integral representation, complete monotonicity, and inequalities of the Catalan numbers

Filomat ◽  
2018 ◽  
Vol 32 (2) ◽  
pp. 575-587 ◽  
Author(s):  
Feng Qi ◽  
Xiao-Ting Shi ◽  
Fang-Fang Liu
Keyword(s):  
Integral Representation ◽  
Generating Function ◽  
Integral Formula ◽  
Cauchy Integral Formula ◽  
Catalan Numbers ◽  
Complete Monotonicity ◽  
Cauchy Integral ◽  
Complex Functions

In the paper, by virtue of the Cauchy integral formula in the theory of complex functions, the authors establish an integral representation for the generating function of the Catalan numbers in combinatorics. From this, the authors derive an alternative integral representation, complete monotonicity, determinantal and product inequalities for the Catalan numbers.

scholarly journals The second-order self-associated orthogonal sequences

2004 ◽  
Vol 2004 (2) ◽  
pp. 137-167 ◽  
Author(s):  
P. Maroni ◽  
M. Ihsen Tounsi
Keyword(s):  
Differential Equation ◽  
Orthogonal Polynomials ◽  
Order Differential Equation ◽  
Second Order ◽  
Integral Representations ◽  
Second Order Differential Equation ◽  
Structure Relation ◽  
Orthogonal Sequences

The aim of this work is to describe the orthogonal polynomials sequences which are identical to their second associated sequence. The resulting polynomials are semiclassical of classs≤3. The characteristic elements of the structure relation and of the second-order differential equation are given explicitly. Integral representations of the corresponding forms are also given. A striking particular case is the case of the so-called electrospheric polynomials.

The Lp-Norm-Minimization Design of Stable Variable-Bandwidth Digital Filters

2018 ◽  
Vol 27 (07) ◽  
pp. 1850102 ◽  
Author(s):  
Tian-Bo Deng
Keyword(s):  
Weighted Least Squares ◽  
Second Order ◽  
Conversion Function ◽  
Magnitude Response ◽  
Lp Norm ◽  
Norm Minimization ◽  
Variable Bandwidth ◽  
The Stability ◽  
Special Case ◽  
Stable Variable

This paper proposes a two-step strategy for designing a variable-bandwidth (VBW) digital filter through minimizing the [Formula: see text]-norm of the magnitude-response error. This [Formula: see text]-norm design can be regarded as a generalized version of the existing weighted-least-squares (WLS) design. Equivalently, the WLS design is a special case of the [Formula: see text]-norm-minimization design for [Formula: see text]. This paper discusses the design of the recursive VBW filter with the transfer function whose denominator is expressed as the product of the second-order sections. As long as all the second-order sections are stable, the recursive VBW filter is also stable. To ensure that the designed recursive VBW filter is stable, we adopt the coefficient-conversion strategy that constrains all the denominator-parameter pairs of the second-order sections within the stability triangle. This paper also proposes a novel conversion function for performing the coefficient conversion. As a consequence, the designed VBW filter is definitely stable. A bandpass VBW filter is designed for showing the feasibility of the [Formula: see text]-norm-minimization-based design and verifying the stability guarantee.

Dependence of the Depolarization Tensor on the Displacement Gradient

1971 ◽  
Vol 49 (17) ◽  
pp. 2284-2286
Author(s):  
A. Redlack ◽  
J. Grindlay
Keyword(s):  
Second Order ◽  
Displacement Gradient ◽  
Derivatives Of ◽  
Special Case

Expressions are given for the first- and second-order derivatives of the depolarization tensor with respect to the displacement gradient. The results are applied to the special case of the sphere.

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