Bell triangle

Abstract: Let [An,k ] n,k ⩾0 be an infinite lower triangular array satisfying the recurrence for n ⩾ 1 and k ⩾ 0, where A 0,0 = 1, A 0,k = A k,–1 = 0 for k > 0. We present some criteria for the log-concavity of rows and strong q-log-convexity of generating functions of rows. Our results can be applied to many well-known triangular arrays, such as the Pascal triangle, the Stirling triangle of the second kind, the Bell triangle, the large Schroder triangle, the Motzkin triangle, and the Catalan triangles of Aigner and Shapiro, in a unified approach. In addition, we prove that the binomial transformation not only preserves the strong q-log-convexity property, but also preserves the strong q-log-concavity property. Finally, we demonstrate that the strong q-log-convexity property is preserved by the Stirling transformation and Whitney transformation of the second kind, which extends some known results for the strong q-log-convexity property.

Abstract: The cryptology is consisted of Kryptos ( hidden) and logos (word) terms in the Greek. It also means that “secrecy science” at the communication. In the present days, the expansion of the electronic comminucation network has more increased the importance of cryptology. In this work, we have focused on Knapsack cryptosystem. In this purpose, the Bell numbers in the form of ‘super- increasing sequence’, which constitute the hypotenuse of the Bell triangle, are generated in the Python programming. The Knapsack encryption and decryption of these numbers are modeled using the Python program. Furthermore, the encryption was tested on a physical equation and it was observed that the Bell numbers are suitable for Knapsack encryption.