Sharadchandra Shankar Shrikhande was an Indian mathematician known for his contributions to various fields in mathematics, particularly in combinatorial designs and finite geometries. He is often recognized for his work on the existence of certain combinatorial structures and the development of the Shrikhande graph, which is a specific graph in graph theory that exhibits interesting properties related to symmetry and structure.

The digamma function, denoted as \( \psi(x) \), is the logarithmic derivative of the gamma function \( \Gamma(x) \). Mathematically, it is defined as: \[ \psi(x) = \frac{d}{dx} \ln(\Gamma(x)) = \frac{\Gamma'(x)}{\Gamma(x)} \] where \( \Gamma'(x) \) is the derivative of the gamma function.

The Gamma function, denoted as \( \Gamma(n) \), is a mathematical function that generalizes the factorial function to complex and real number arguments. For any positive integer \( n \), the Gamma function satisfies the relation: \[ \Gamma(n) = (n-1)! \] The Gamma function is defined for all complex numbers except for the non-positive integers.

The K-function, or K statistic, is a tool used in spatial statistics to analyze the distribution of points in a given space. It is particularly useful in evaluating whether the spatial pattern of points in a dataset is clustered, random, or dispersed. The K-function is defined for a specific radius \( r \) and is calculated as follows: 1. For each point in the dataset, determine how many other points lie within a distance \( r \).

The Nu function is not a standard mathematical or scientific function widely recognized in literature or academia. However, if you are referring to a function or concept that is known by a specific name or acronym, please provide more context.

Stirling's approximation is a formula used to approximate the factorial of a large integer \( n \). It is particularly useful in combinatorics, statistical mechanics, and various areas of mathematics and physics where factorials of large numbers arise. The approximation is given by the formula: \[ n!

The trigamma function, denoted as \(\psi' (x)\) or sometimes as \(\mathrm{Trigamma}(x)\), is the derivative of the digamma function \(\psi(x)\), which is itself the logarithmic derivative of the gamma function \(\Gamma(x)\).

Perles configuration refers to a specific arrangement in set theory and combinatorial geometry related to the study of convex sets in Euclidean spaces. Named after the mathematician R. Perles, this configuration typically consists of a set of points in general position (no three points are collinear) and relates to properties such as convex hulls and the combinatorial aspects of point sets.

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles, particularly right-angled triangles. The word "trigonometry" comes from the Greek words "trigonon" (triangle) and "metron" (measure).

The Soboleva modified hyperbolic tangent function, often represented as \( \tanh_s(x) \), is a mathematical function that is a modification of the standard hyperbolic tangent function. In various domains, including physics and engineering, such modified functions are introduced to better handle specific properties such as asymptotic behavior, smoothness, or to meet certain boundary conditions.

Carlson symmetric form is a mathematical representation used primarily in the context of complex analysis and number theory, particularly in the theory of modular forms and elliptic functions. It is named after the mathematician Borchardt Carlson. In simple terms, the Carlson symmetric form is a way to express certain types of functions that are symmetric in their arguments.

Dixon elliptic functions are a set of functions that arise in the theory of elliptic functions, which are complex functions that are periodic in two different directions. Specifically, Dixon elliptic functions are a generalization of the classical elliptic functions and are studied primarily in the context of algebraic functions and complex analysis. Named after the mathematician Alfred William Dixon, these functions have particular properties that make them useful in various branches of mathematics, including number theory, algebraic geometry, and mathematical physics.

The half-period ratio, often referred to in the context of periodic functions, is a mathematical concept that describes the relationship between the periods of a function and its symmetry properties. Specifically, for a periodic function, the half-period ratio relates the half-period to the full period of the function. More formally, if \( T \) is the full period of a periodic function, then the half-period, denoted as \( T/2 \), is simply half of that period.

The bilateral hypergeometric series is a generalization of the ordinary hypergeometric series, which allows for the summation of terms indexed by two parameters rather than one.

The Frobenius solution to the hypergeometric equation refers to the method of finding a series solution near a regular singular point of the hypergeometric differential equation.

The Legendre functions, often referred to in the context of Legendre polynomials and Legendre functions of the first and second kind, arise in the solution of a variety of problems in physics and engineering, particularly in the fields of potential theory and solving partial differential equations. 1. **Legendre Polynomials**: These are a sequence of orthogonal polynomials defined on the interval \([-1, 1]\) and are denoted as \(P_n(x)\).

Schwarz's list is a classification of certain interesting or notable groups of mathematical objects, specifically in the context of algebraic topology and complex geometry. It is named after the mathematician Hermann Schwarz. In algebraic topology, Schwarz's list typically refers to specific examples or types of manifolds that exhibit particular properties or behaviors, often with an emphasis on those that are closely related to the study of Riemann surfaces, complex manifolds, or other geometric structures.

The **chromatic polynomial** is a polynomial that encodes the number of ways to color the vertices of a graph using a given number of colors such that no two adjacent vertices share the same color. Specifically, for a graph \( G \), the chromatic polynomial \( P(G, k) \) is defined as the number of ways to color the graph with \( k \) colors. ### Key Properties 1.

Star coloring is a type of graph coloring in which the vertices of a graph are assigned colors such that no two adjacent vertices share the same color and additionally, no two vertices at a distance of two (i.e., connected through a single vertex) have the same color.

The De Bruijn–Erdős theorem is a result in graph theory concerning the existence of graphs with certain properties. Specifically, it addresses the conditions under which a graph can be constructed with a prescribed degree sequence and certain independence properties.