This contribution proposes a novel steganographic method based on the compression standard according to the Joint Photographic Expert Group and an Entropy Thresholding technique. The steganographic algorithm uses one public key and one private key to generate a binary sequence of pseudorandom numbers that indicate where the elements of the binary sequence of a secret message will be inserted. The insertion takes eventually place at the first seven AC coefficients in the transformed DCT domain. Before the insertion of the message the image undergoes several transformations. After the insertion the inverse transformations are applied in reverse order to the original transformations. The insertion itself takes only place if an entropy threshold of the corresponding block is satisfied and if the pseudorandom number indicates to do so. The experimental work on the validation of the algorithm consists of the calculation of the peak signal-to-noise ratio (PSNR), the difference and correlation distortion metrics, the histogram analysis, and the relative entropy, comparing the same characteristics for the cover and stego image. The proposed algorithm improves the level of imperceptibility analyzed through the PSNR values. A steganalysis experiment shows that the proposed algorithm is highly resistant against the Chi-square attack.

In this contribution we consider sequences of monic polynomials orthogonal with respect to a Sobolev-type inner productwhere u is a q-classical linear functional and D q is the q-derivative operator. We obtain some algebraic properties of these polynomials such as an explicit representation, a five-term recurrence relation as well as a second order linear q-difference holonomic equation fulfilled by such polynomials.We present an analysis of the behaviour of its zeros as a function of the mass N . In particular, we obtain the exact values of N such that the smallest (respectively, the greatest) zero of the studied polynomials is located outside of the support of the measure. We conclude this work considering two examples.

In this contribution we consider the sequence {Q λ n } n≥0 of monic polynomials orthogonal with respect to the following inner product involving differenceswhere λ ∈ R + , ∆ denotes the forward difference operator defined by ∆f (x) = f (x + 1) − f (x), ψ (a) with a > 0 is the well known Poisson distribution of probability theory dψ (a) (x) = e −a a x x! at x = 0, 1, 2, . . . , and c ∈ R is such that ψ (a) has no points of increase in the interval (c, c + 1). We derive its corresponding hypergeometric representation. The ladder operators and two different versions of the linear difference equation of second order corresponding to these polynomials are given. Recurrence formulas of five and three terms, the latter with rational coefficients, are presented. Moreover, for real values of c such that c + 1 < 0, we obtain some results on the distribution of its zeros as decreasing functions of λ, when this parameter goes from zero to infinity.

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This contribution deals with the sequence {Un(a)(x;q,j)}n≥0 of monic polynomials in x, orthogonal with respect to a Sobolev-type inner product related to the Al-Salam–Carlitz I orthogonal polynomials, and involving an arbitrary number j of q-derivatives on the two boundaries of the corresponding orthogonality interval, for some fixed real number q∈(0,1). We provide several versions of the corresponding connection formulas, ladder operators, and several versions of the second order q-difference equations satisfied by polynomials in this sequence. As a novel contribution to the literature, we provide certain three term recurrence formula with rational coefficients satisfied by Un(a)(x;q,j), which paves the way to establish an appealing generalization of the so-called J-fractions to the framework of Sobolev-type orthogonality.

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New (infinitely many) rational approximants to ζ(3) proving its irrationality are given. The recurrence relations for the numerator and denominator of these approximants as well as their continued fraction expansions are obtained. A comparison of our approximants with Apéry's approximants to ζ(3) is shown.

In this contribution, we consider the sequence {Hn(x;q)}n≥0 of monic polynomials orthogonal with respect to a Sobolev-type inner product involving forward difference operators For the first time in the literature, we apply the non-standard properties of {Hn(x;q)}n≥0 in a watermarking problem. Several differences are found in this watermarking application for the non-standard cases (when j>0) with respect to the standard classical Krawtchouk case λ=μ=0.