Generalized Pascal triangles and pyramids, their fractals, graphs and applications. (Obobshchennye treugol’niki i piramidy Paskalya, ikh fraktali, grafy i prilozheniya.) (Russian) Zbl 0706.05002

Bondarenko, B. A.

Tashkent: Fan. 192 p. R. 2.10 (1990).

The known results connected with Pascal’s triangles and their generalizations are systematized. In contrast to the excellent popular books of V. A. Uspenskij [Pascal’s triangle (Moscow Mir 1979) (Chicago 1974; Zbl 0318.05001)] and T. M. Green and C. L. Hamberg [Pascal’s triangle (1986)] this monograph deals with deeper questions. It cosiders the problems of divisibility for binomial and generally polynomial coefficients, their distributions (mod \(p^ s)\) in the corresponding arithmetic triangles, pyramids and hyperpyramids (p is a prime numbers, \(s\in {\mathbb{Z}})\). The author especially studies fractals obtained on the basis of Pascal’s and other arithmetic triangles.
The book contains a lot of applications to combinatorial analysis and the theory of numbers, mathematical physics and computer sciences. It consists of the following chapters:
1. Pascal’s triangle, its plane and space generalizations. 2. Divisibility and distribution (mod \(p^ s)\) of binomial, trinomial and polynomial coefficients. 3. Divisibility and distribution (mod p) of generalized binomial coefficients and Fibonacci, Lucas etc. sequences.
4. Fractals of Pascal’s triangle and of other arithmetic triangles.
5. Generalized arithmetic graphs and their properties.
6. Matrices and determinants composed of binomial, generalized binomial coefficients and other numbers.
7. Combinatorial algorithms for the construction of generalized homogeneous polynomials. Some classes of nonorthogonal polynomials.
Bibliography contains more than 400 references.

Reviewer: A.S.Slavutskij

MSC:

05A10	Factorials, binomial coefficients, combinatorial functions
05-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to combinatorics
11-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory
11B65	Binomial coefficients; factorials; \(q\)-identities
11B39	Fibonacci and Lucas numbers and polynomials and generalizations

Keywords:

Pascal’s triangles

Citations:

Zbl 0318.05001