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3x+1 inverse orbit generating functions almost always have natural boundaries

Jason P. Bell, Jeffrey C. Lagarias (2015)

Acta Arithmetica

The 3x+k function T k ( n ) sends n to (3n+k)/2, resp. n/2, according as n is odd, resp. even, where k ≡ ±1 (mod 6). The map T k ( · ) sends integers to integers; for m ≥1 let n → m mean that m is in the forward orbit of n under iteration of T k ( · ) . We consider the generating functions f k , m ( z ) = n > 0 , n m z n , which are holomorphic in the unit disk. We give sufficient conditions on (k,m) for the functions f k , m ( z ) to have the unit circle |z|=1 as a natural boundary to analytic continuation. For the 3x+1 function these conditions hold for all m...


Е.М. Никишин (1980)

Matematiceskij sbornik

A Dichotomy Principle for Universal Series

V. Farmaki, V. Nestoridis (2008)

Bulletin of the Polish Academy of Sciences. Mathematics

Applying results of the infinitary Ramsey theory, namely the dichotomy principle of Galvin-Prikry, we show that for every sequence ( α j ) j = 1 of scalars, there exists a subsequence ( α k j ) j = 1 such that either every subsequence of ( α k j ) j = 1 defines a universal series, or no subsequence of ( α k j ) j = 1 defines a universal series. In particular examples we decide which of the two cases holds.

A majorant problem.

Peretz, Ronen (1992)

International Journal of Mathematics and Mathematical Sciences

A note on the number of zeros of polynomials in an annulus

Xiangdong Yang, Caifeng Yi, Jin Tu (2011)

Annales Polonici Mathematici

Let p(z) be a polynomial of the form p ( z ) = j = 0 n a j z j , a j - 1 , 1 . We discuss a sufficient condition for the existence of zeros of p(z) in an annulus z ∈ ℂ: 1 - c < |z| < 1 + c, where c > 0 is an absolute constant. This condition is a combination of Carleman’s formula and Jensen’s formula, which is a new approach in the study of zeros of polynomials.

A recovery of Brouncker's proof for the quadrature continued fraction.

Sergey Khrushchev (2006)

Publicacions Matemàtiques

350 years ago in Spring of 1655 Sir William Brouncker on a request by John Wallis obtained a beautiful continued fraction for 4/π. Brouncker never published his proof. Many sources on the history of Mathematics claim that this proof was lost forever. In this paper we recover the original proof from Wallis' remarks presented in his Arithmetica Infinitorum. We show that Brouncker's and Wallis' formulas can be extended to MacLaurin's sinusoidal spirals via related Euler's products. We derive Ramanujan's...

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