Binomial transform

Abstract: Extending earlier work of R. Donaghey and P. J. Cameron, we investigate some canonical "eigen-sequences" associated with transformations of integer sequences. Several known sequences appear in a new setting: for instance the sequences (such as 1, 3, 11, 49, 257, 1531, ...) studied by T. Tsuzuku, H. O. Foulkes and A. Kerber in connection with multiply transitive groups are eigen-sequences for the binomial transform. Many interesting new sequences also arise, such as 1, 1, 2, 26, 152, 1144, ..., which shifts one place left when transformed by the Stirling numbers of the second kind, and whose exponential generating function satisfies A'(x) = A(e^x -1) + 1.

Abstract: In this paper the finite Hankel transformation of generalized function of a certain space is defined, and an inversion formula for the transformation is established. The inversion formula gives rise to a Fourier-Bessel series expansion of generalized functions. The convergence of the series is interpreted in the weak distributional sense. An operation transform formula is also obtained, which together with the inversion formula, is applied in solving certain distributional differential equations.

Abstract: Multiple binomial sums form a large class of multi-indexed sequences, closed under partial summation, which contains most of the sequences obtained by multiple summation of products of binomial coefficients and also all the sequences with algebraic generating function. We study the representation of the generating functions of binomial sums by integrals of rational functions. The outcome is twofold. Firstly, we show that a univariate sequence is a multiple binomial sum if and only if its generating function is the diagonal of a rational function. Secondly, we propose algorithms that decide the equality of multiple binomial sums and that compute recurrence relations for them. In conjunction with geometric simplifications of the integral representations, this approach behaves well in practice. The process avoids the computation of certificates and the problem of the appearance of spurious singularities that afflicts discrete creative telescoping, both in theory and in practice.

Abstract: In this paper, a novel Binomial transform based fragile watermarking technique has been proposed for color image authentication. Binomial transform (BT) is applied to convert each 2ź×ź2 sub-image block into transform domain in row major order. On average, two bits of authenticating watermark are fabricated on each transformed component starting from the least significant bit position (LSB-0). Inverse Binomial transform (IBT) is performed as post-embedding operation to convert each 2ź×ź2 transformed block back into the spatial domain. A delicate re-adjustment is performed on the first embedded component to keep the pixel components positive and less than or equal to 255 keeping the fabricated watermark unaltered. The watermark is extracted at the recipient end based on the reverse operation and is verified for authentication using a message digest. Experimental result ensures that the proposed technique obtain higher Payload and Peak Signal to Noise Ratio (PSNR) as compared to existing methods.