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Narayana Polynomials
Narayana polynomials are a class of polynomials whose coefficients are the Narayana numbers. The Narayana numbers and Narayana polynomials are named after the Canadian mathematician T. V. Narayana (1930–1987). They appear in several combinatorial problems. Definitions For a positive integer n and for an integer k\geq0, the Narayana number N(n,k) is defined by : N(n,k) = \frac. The number N(0,k) is defined as 1 for k=0 and as 0 for k\ne0. For a nonnegative integer n , the n-th Narayana polynomial N_n(z) is defined by :N_n(z) = \sum_^n N(n,k)z^k. The associated Narayana polynomial \mathcal N_n(z) is defined as the reciprocal polynomial of N_n(z): :\mathcal N_n(z)=z^nN_n\left(\tfrac\right). Examples The first few Narayana polynomials are :N_0(z)=1 :N_1(z)=z :N_2(z)=z^2+z :N_3(z)=z^3+3z^2+z :N_4(z)=z^4+6z^3+6z^2+z :N_5(z)=z^5+10z^4+20z^3+10z^2+z Properties A few of the properties of the Narayana polynomials and the associated Narayana polynomials are collected below. Furt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Narayana Numbers
Narayana (, ) is one of the forms and epithets of Vishnu. In this form, the deity is depicted in yogic slumber under the Kshira Sagara, celestial waters, symbolising the masculine principle and associated with his role of creation. He is also known as Purushottama, and is considered the Brahman, Supreme Being in Vaishnavism. Etymology Narayan Aiyangar states the meaning of the Sanskrit word 'Narayana' can be traced back to the Manusmriti, Laws of Manu (also known as the ''Manusmriti'', a ''Dharmaśāstra'' text), which states: This definition is used throughout post-Vedic literature such as the ''Mahabharata'' and the ''Vishnu Purana''. 'Narayana' is also defined as the 'son of the Purusha, primeval man', and 'Supreme Being who is the foundation of all men'. *'Nara' (Sanskrit नार) means 'water' and 'man' *'Yana' (Sanskrit यान) means 'vehicle', 'vessel', or more loosely, 'abode' or 'home' L. B. Keny proposes that Narayana was associated with the Dravidian pe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tadepalli Venkata Narayana
T. V. Narayana (Tadepalli Venkata Narayana) (23 April 1930 – 6 February 1987) was an Indo-Canadian mathematical statistician and mathematician known for his contributions to combinatorics, lattice theory and mathematical statistics. A certain sequence of numbers called the Narayana numbers and a certain class of polynomials called Narayana polynomials, which were both named after him for his work in bringing out their importance in lattice path theory, have found extensive and varied applications in combinatorics and lattice theory. (Published by the Statistical Society of Canada.) Narayana was born in Madras (now Chennai), India on 23 April 1930. He studied at the Madras and Bombay universities in India before joining the North Carolina University for pursuing PhD studies under the supervision of Raj Chandra Bose. He was awarded the PhD degree in mathematical statistics in the year 1954 for a dissertation titled ''Sequential Procedures in Probit Analysis''. After securing PhD, Na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reciprocal Polynomial
In algebra, given a polynomial :p(x) = a_0 + a_1x + a_2x^2 + \cdots + a_nx^n, with coefficients from an arbitrary field, its reciprocal polynomial or reflected polynomial,* denoted by or , is the polynomial :p^*(x) = a_n + a_x + \cdots + a_0x^n = x^n p(x^). That is, the coefficients of are the coefficients of in reverse order. Reciprocal polynomials arise naturally in linear algebra as the characteristic polynomial of the inverse of a matrix. In the special case where the field is the complex numbers, when :p(z) = a_0 + a_1z + a_2z^2 + \cdots + a_nz^n, the conjugate reciprocal polynomial, denoted , is defined by, :p^(z) = \overline + \overlinez + \cdots + \overlinez^n = z^n\overline, where \overline denotes the complex conjugate of a_i, and is also called the reciprocal polynomial when no confusion can arise. A polynomial is called self-reciprocal or palindromic if . The coefficients of a self-reciprocal polynomial satisfy for all . Properties Reciprocal polynomial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Catalan Number
The Catalan numbers are a sequence of natural numbers that occur in various Enumeration, counting problems, often involving recursion, recursively defined objects. They are named after Eugène Charles Catalan, Eugène Catalan, though they were previously discovered in the 1730s by Minggatu. The -th Catalan number can be expressed directly in terms of the central binomial coefficients by :C_n = \frac = \frac \qquad\textn\ge 0. The first Catalan numbers for are : . Properties An alternative expression for is :C_n = - for n\ge 0\,, which is equivalent to the expression given above because \tbinom=\tfrac\tbinomn. This expression shows that is an integer, which is not immediately obvious from the first formula given. This expression forms the basis for a #Second proof, proof of the correctness of the formula. Another alternative expression is :C_n = \frac \,, which can be directly interpreted in terms of the cycle lemma; see below. The Catalan numbers satisfy the recurr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schröder Number
In mathematics, the Schröder number S_n, also called a ''large Schröder number'' or ''big Schröder number'', describes the number of lattice paths from the southwest corner (0,0) of an n \times n grid to the northeast corner (n,n), using only single steps north, (0,1); northeast, (1,1); or east, (1,0), that do not rise above the SW–NE diagonal. The first few Schröder numbers are :1, 2, 6, 22, 90, 394, 1806, 8558, ... . where S_0=1 and S_1=2. They were named after the German mathematician Ernst Schröder (mathematician), Ernst Schröder. Examples The following figure shows the 6 such paths through a 2 \times 2 grid: Related constructions A Schröder path of length n is a lattice path from (0,0) to (2n,0) with steps northeast, (1,1); east, (2,0); and southeast, (1,-1), that do not go below the x-axis. The nth Schröder number is the number of Schröder paths of length n. The following figure shows the 6 Schröder paths of length 2. Similarly, the Schröder numbers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generating Function
In mathematics, a generating function is a representation of an infinite sequence of numbers as the coefficients of a formal power series. Generating functions are often expressed in closed form (rather than as a series), by some expression involving operations on the formal series. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series. Every sequence in principle has a generating function of each type (except that Lambert and Dirichlet series require indices to start at 1 rather than 0), but the ease with which they can be handled may differ considerably. The particular generating function, if any, that is most useful in a given context will depend upon the nature of the sequence and the details of the problem being addressed. Generating functions are sometimes called generating series, in that a series of terms can be said to be the generator of its sequence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Legendre Polynomial
In mathematics, Legendre polynomials, named after Adrien-Marie Legendre (1782), are a system of complete and orthogonal polynomials with a wide number of mathematical properties and numerous applications. They can be defined in many ways, and the various definitions highlight different aspects as well as suggest generalizations and connections to different mathematical structures and physical and numerical applications. Closely related to the Legendre polynomials are associated Legendre polynomials, Legendre functions, Legendre functions of the second kind, big q-Legendre polynomials, and associated Legendre functions. Definition and representation Definition by construction as an orthogonal system In this approach, the polynomials are defined as an orthogonal system with respect to the weight function w(x) = 1 over the interval 1,1/math>. That is, P_n(x) is a polynomial of degree n, such that \int_^1 P_m(x) P_n(x) \,dx = 0 \quad \text n \ne m. With the additional sta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
