Transcendental values of the incomplete gamma function and related questions

@article{Murty2015TranscendentalVO,
  title={Transcendental values of the incomplete gamma function and related questions},
  author={M. Murty and Ekata Saha},
  journal={Archiv der Mathematik},
  year={2015},
  volume={105},
  pages={271-283}
}
For s, x > 0, the lower incomplete gamma function is defined to be the integral $${\gamma(s,x):=\int_{0}^{x} t^{s} e^{-t} \frac{dt}{t}}$$γ(s,x):=∫0xtse-tdtt, which can be continued analytically to an open subset of $${\mathbb{C}^{2}}$$C2. Here in this article, we study the transcendence of special values of the lower incomplete gamma function, by means of transcendence of certain infinite series. These series are variants of series which are of great interest in number theory. However, these… Expand
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References

SHOWING 1-10 OF 22 REFERENCES
Special values of the Gamma function at CM points
Little is known about the transcendence of certain values of the Gamma function, Γ(z). In this article, we study values of Γ(z) when $\mathbb{Q}(z)$ is an imaginary quadratic field. We also studyExpand
Infinite sums as linear combinations of polygamma functions
where P (x), Q(x) are polynomials with algebraic coefficients satisfying some symmetry conditions, f is a number theoretic periodic function taking algebraic values, and all the roots of theExpand
Transcendental values of the digamma function
Let ψ(x) denote the digamma function, that is, the logarithmic derivative of Euler's Γ-function. Let q be a positive integer greater than 1 and γ denote Euler's constant. We show that all the numbersExpand
Measures of simultaneous approximation for quasi-periods of abelian varieties
Abstract We examine various extensions of a series of theorems proved by Chudnovsky in the 1980s on the algebraic independence (transcendence degree 2) of certain quantities involving integrals ofExpand
TRANSCENDENTAL NUMBERS
The Greeks tried unsuccessfully to square the circle with a compass and straightedge. In the 19th century, Lindemann showed that this is impossible by demonstrating that π is not a root of anyExpand
Unearthing the visions of a master: harmonic Maass forms and number theory
Together with his collaborators, most notably Kathrin Bringmann and Jan Bruinier, the author has been researching harmonic Maass forms. These non-holomorphic modular forms play central roles in manyExpand
Applications of a theorem by A. B. Shidlovski
  • K. Mahler
  • Mathematics
  • Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences
  • 1968
Shidlovski’s deep theorem on Siegel E-functions satisfying systems of linear differential equations is applied in this paper to the study of the arithmetic properties of the partial derivatives Ck(z)Expand
Differential operators for harmonic weak Maass forms and the vanishing of Hecke eigenvalues
For integers k ≥ 2, we study two differential operators on harmonic weak Maass forms of weight 2 − k. The operator ξ2-k (resp. Dk-1) defines a map to the space of weight k cusp forms (resp. weaklyExpand
A refined version of the Siegel-Shidlovskii theorem
Using Y.Andr\'e's result on differential equations staisfied by $E$-functions, we derive an improved version of the Siegel-Shidlovskii theorem. It gives a complete characterisation of algebraicExpand
Transcendental infinite sums
Abstract We show that it follows from results on linear forms in logarithms of algebraic numbers such as where χ is any non-principal Dirichlet character and (Fn∞n=0 the Fibonacci sequence, areExpand
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