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{\Large{\bf{Oral Exam}}} \\
Thotsaporn Thanatipanonda \\
thot@math.rutgers.edu  
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{\large{\bf{Enumerative Combinatorics}}} 

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{\bf{Basic enumeration}}: recurrence relations, inclusion-exclusion
principle, permutation statistics, Stirling numbers, Bell numbers,
calculus of finite differences.  \\

{\bf{Generating functions}}: formal power series, formal exponential
generating functions, Dirichlet series, moments of distributions, cycle
index of the symmetric group, snake oil method, M\"{o}bius inversion,
Lagrange inversion.  \\

{\bf{The five basic algorithms for hypergeometric identities}}: Mary
Celine Fasenmyer's algorithm, Gosper's algorithm, Zeilberger's algorithm,
Petkov\v{s}ek's algorithm, the WZ phenomena, dual and companion 
identities. \\

{\bf{Rational generating functions}}: rational power series in one
variable, transfer-matrix method, factorization in free monoids. \\

{\bf{Computer algebra}}: Maple programming. \\

{\noindent {\bf References}}

\noindent Richard Stanley; {\em{Enumerative Combinatorics, Vol. 1}},
Chapters 1,4. \\
Herbert Wilf; {\em{Generatingfunctionology}}, Chapters 1,2,4. \\ 
Marko Petkov\v{s}ek, Herbert Wilf, Doron Zeilberger; {\em{A=B}}, Chapters 
1-8. \\

{\large{\bf{Special Functions}}} 

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{\bf{The gamma and beta functions}}: definition, Euler's reflection
formula, the Hurwitz and Riemann zeta functions, Stirling's asymptotic
formula, Gauss's multiplication formula for the gamma function,
Bohr-Mollerup theorem, probabilistic evaluation of
the beta function. \\

{\bf{Hypergeometric functions}}: definition, Euler's integral
representation, Gauss formula,  contiguous 
relation, 
dilogarithms, binomial sums, Dougall's bilateral sum. \\ 

{\bf{General Theory of Orthogonal polynomials}}:  three-term recurrences,
Gauss quadrature, zeros of orthogonal polynomials, continued fractions,
kernel polynomials, Parseval's formula.  \\

{\bf{Classical and Special Orthogonal polynomials}}: Hermite polynomials,
Laguerre polynomials, Chebyshev polynomials, Jacobi polynomials, An
extension of the Ultraspherical Polynomials, Wilson polynomials \\

{\bf{q-Series}}: q-Binomial theorem, Jacobi triple product formula, 
Ramanujan's Summation Formula, Basic Hypergeometric series, 
q-Ultraspherical polynomials. \\

{\bf {Askey Scheme}}: hypergeometic orthogonal polynomials, Askey-Wilson,
Continuous dual Hahn, Jacobi, Meixner, Laguerre, Hermite and their
q-analogs. \\
 
{\bf{Quasideterminants}}: definition, general properties, basic 
identities. \\

{\noindent {\bf References}}

\noindent George Andrews, Richard Askey, Ranjan Roy; {\em {Special 
Functions}}, Chapters 1,2,5,6,10. \\
I.M. Gelfand, V. Retakh; {\em{Quasideterminants}}, Selecta Mathematica 
3 (1997) \\

{\large{\bf{Graph Theory}}} 

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{\bf{Basic graph theory}}: trees, bipartite graphs, Eulerian tour, 
path and cycles. \\

{\bf{Matchings}}: K\"{o}nig's theorem, Hall's theorem, Tutte's theorem, 
path covers and Dilworth's theorem. \\

{\bf{Connectivity}}: structure of 2-connected and 3-connected graphs,
Menger's theorem, Mader's theorem, linking pairs of vertices.  \\

{\bf{Planar graphs}}: Euler's formula, Kuratowski's theorem, planar 
duality. \\

{\bf{Random graphs}}: Erd\H{o}s's theorem, probabilistic method, threshold 
functions, second moments. \\

{\noindent {\bf References}}

\noindent Reinhard Diestel; {\em{Graph Theory}}, Chapters 1-4, 11.       

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