By Ira M. Gessel, Sangwook Ree (auth.), N. Balakrishnan (eds.)

Sri Gopal Mohanty has made pioneering contributions to lattice course counting and its functions to likelihood and information. this can be sincerely glaring from his lifetime guides checklist and the various citations his courses have got over the last 3 a long time. My organization with him all started in 1982 whilst I got here to McMaster Univer sity. due to the fact that then, i've been linked to him on many alternative concerns at expert in addition to cultural degrees; i've got benefited significantly from him on either those grounds. i've got loved a great deal being his colleague within the information workforce right here at McMaster college and in addition as his buddy. whereas i love him for his honesty, sincerity and commitment, I take pleasure in greatly his kindness, modesty and broad-mindedness. apart from our universal curiosity in arithmetic and data, we either have nice love for Indian classical track and dance. now we have spent various many various matters linked to the Indian tune and hours discussing dance. I nonetheless bear in mind fondly the lengthy force (to Amherst, Massachusetts) I had many years in the past with him and his spouse, Shantimayee, and all of the hearty discussions we had in the course of that trip. Combinatorics and purposes of combinatorial tools in likelihood and records has turn into a truly lively and fertile region of study within the contemporary past.

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**Sample text**

_ bn-i(x-ai-c)+ ~ Yi _. _ bn_i(x-ai-c). 6) finishes the expansion, but makes it into a double sum. This procedure can be repeated for initial values on more affine pieces. Obviously, the multiplicity of the summation will grow with the number of pieces. 78994x2 + 4. 2212x - 1. 3 + 2 tan 1 tan 3) n + 4 tan 3 - xn (tan 2 - tan 1) n tan 2 Xn (tan 4 - 3 tan 4 for n = 0, 1, 2 for n ~ 3. Applications: Bounded paths Some of the best known applications occur in the enumeration of lattice paths, sequences of horizontal ---+ and vertical Tsteps starting at the origin.

The linear combination sn(x) := bn(x - c) - aPn-I(:r - c) = x - an x-c C bn(x - c) of Sheffer polynomials is again a Sheffer polynomial for the same operator, and solves the initial value BS n = Sn-I, and sn(an + c) = 80,n for all n = 0,1, ... , where a and c are given constants. This solution has already been given in Rota, Kahaner and Odlyzko (1973). In order to solve the problem BTn = Tn-I, and Tn(an + c) = Yn for all n = 0, 1, ... 3), we must define t~)(x) := sn(x-ai) and get n n Tn(X) = "~ YiSn-i(X - ai) = "~ Yi x-an-c .

Niederhausen, H. (1986). The enumeration ofrestricted random walks by Sheffer polynomials with applications to statistics, Journal of Statistical Planning and Inference, 14, 95-114. 12. Niederhausen, H. (1992). Fast Lagrange inversion, with an application to factorial numbers, Discrete Mathematics, 104, 99-110. 13. Niederhausen, H. (1994). Counting intersecting weighted pairs of lattice paths using transforms of operators, Congressus Numerantium, 102, 161173. 14. Niederhausen, H. (1996). Symmetric Sheffer sequences and their applications to lattice path counting, Journal of Statistical Planning and Inference, 54, 87-100.