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11: 27.13 Functions
The basic problem is that of expressing a given positive integer n as a sum of integers from some prescribed set S whose members are primes, squares, cubes, or other special integers. … This problem is named after Edward Waring who, in 1770, stated without proof and with limited numerical evidence, that every positive integer n is the sum of four squares, of nine cubes, of nineteen fourth powers, and so on. …
27.13.5 ( ϑ ( x ) ) 2 = 1 + n = 1 r 2 ( n ) x n .
By similar methods Jacobi proved that r 4 ( n ) = 8 σ 1 ( n ) if n is odd, whereas, if n is even, r 4 ( n ) = 24 times the sum of the odd divisors of n . …
12: 26.12 Plane Partitions
where σ 2 ( j ) is the sum of the squares of the divisors of j . …
13: 27.6 Divisor Sums
27.6.1 d | n λ ( d ) = { 1 , n  is a square , 0 , otherwise .
14: 26.18 Counting Techniques
26.18.1 | S ( A 1 A 2 A n ) | = | S | + t = 1 n ( - 1 ) t 1 j 1 < j 2 < < j t n | A j 1 A j 2 A j t | .
26.18.2 N + t = 1 n ( - 1 ) t 1 j 1 < j 2 < < j t n N p j 1 p j 2 p j t .
With the notation of §26.15, the number of placements of n nonattacking rooks on an n × n chessboard that avoid the squares in a specified subset B is
26.18.3 n ! + t = 1 n ( - 1 ) t r t ( B ) ( n - t ) ! .
26.18.4 k n + t = 1 n ( - 1 ) t ( k t ) ( k - t ) n .
15: 28.30 Expansions in Series of Eigenfunctions
Then every continuous 2 π -periodic function f ( x ) whose second derivative is square-integrable over the interval [ 0 , 2 π ] can be expanded in a uniformly and absolutely convergent series
28.30.3 f ( x ) = m = 0 f m w m ( x ) ,
16: 22.11 Fourier and Hyperbolic Series
17: Bibliography
  • T. M. Apostol and H. S. Zuckerman (1951) On magic squares constructed by the uniform step method. Proc. Amer. Math. Soc. 2 (4), pp. 557–565.
  • T. M. Apostol (1952) Theorems on generalized Dedekind sums. Pacific J. Math. 2 (1), pp. 1–9.
  • T. M. Apostol (2006) Bernoulli’s power-sum formulas revisited. Math. Gaz. 90 (518), pp. 276–279.
  • R. Askey and G. Gasper (1976) Positive Jacobi polynomial sums. II. Amer. J. Math. 98 (3), pp. 709–737.
  • R. Askey (1974) Jacobi polynomials. I. New proofs of Koornwinder’s Laplace type integral representation and Bateman’s bilinear sum. SIAM J. Math. Anal. 5, pp. 119–124.
  • 18: 1.8 Fourier Series
    1.8.1 f ( x ) = 1 2 a 0 + n = 1 ( a n cos ( n x ) + b n sin ( n x ) ) ,
    1.8.3 f ( x ) = n = - c n e i n x ,
    Then the series (1.8.1) converges to the sumwhen f ( x ) and g ( x ) are square-integrable and a n , b n and a n , b n are their respective Fourier coefficients. …
    1.8.15 1 2 f ( 0 ) + n = 1 f ( n ) = 0 f ( x ) d x + 2 n = 1 0 f ( x ) cos ( 2 π n x ) d x .
    19: 3.5 Quadrature
    3.5.5 - f ( t ) d t = h k = - f ( k h ) + E h ( f ) ,
    §3.5(x) Cubature Formulas
    and the square S , given by | x | h , | y | h : …
    Table 3.5.21: Cubature formulas for disk and square.
    Diagram ( x j , y j ) w j R
    20: 14.28 Sums
    §14.28 Sums
    14.28.1 P ν ( z 1 z 2 - ( z 1 2 - 1 ) 1 / 2 ( z 2 2 - 1 ) 1 / 2 cos ϕ ) = P ν ( z 1 ) P ν ( z 2 ) + 2 m = 1 ( - 1 ) m Γ ( ν - m + 1 ) Γ ( ν + m + 1 ) P ν m ( z 1 ) P ν m ( z 2 ) cos ( m ϕ ) ,
    where the branches of the square roots have their principal values when z 1 , z 2 ( 1 , ) and are continuous when z 1 , z 2 ( 0 , 1 ] . …
    14.28.2 n = 0 ( 2 n + 1 ) Q n ( z 1 ) P n ( z 2 ) = 1 z 1 - z 2 , z 1 1 , z 2 2 ,
    §14.28(iii) Other Sums