Carlitz Polynomial (other)
In mathematics, Carlitz polynomial, named for Leonard Carlitz, may refer to: *Al-Salam–Carlitz polynomials *Tricomi–Carlitz polynomials In mathematics, the Tricomi–Carlitz polynomials or (Carlitz–)Karlin–McGregor polynomials are polynomials studied by and and , related to random walks on the positive integers. They are given in terms of Laguerre polynomials In mathematics, ...

Leonard Carlitz
Leonard Carlitz (December 26, 1907 – September 17, 1999) was an American mathematician. Carlitz supervised 44 doctorates at Duke University and published over 770 papers. Chronology * 1907 Born Philadelphia, PA, USA * 1927 BA, University of Pennsylvania * 1930 PhD, University of Pennsylvania, 1930 under Howard Mitchell, who had studied under Oswald Veblen at Princeton * 1930–31 at Caltech with E. T. Bell * 1931 married Clara Skaler * 1931–32 at Cambridge with G. H. Hardy * 1932 Joined the faculty of Duke University where he served for 45 years * 1938 to 1973 Editorial Board Duke Mathematical Journal (Managing Editor from 1945.) * 1939 Birth of son Michael * 1940 Supervision of his first doctoral student E. F. Canaday, awarded 1940 * 1945 Birth of son Robert * 1964 First James B. Duke Professor in Mathematics * 1977 Supervised his 44th and last doctoral student, Jo Ann Lutz, awarded 1977 * 1977 Retired * 1990 Death of wife Clara, after 59 years of marriage * 1999 Sep ...
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Al-Salam–Carlitz Polynomials
In mathematics, Al-Salam–Carlitz polynomials ''U''(''x'';''q'') and ''V''(''x'';''q'') are two families of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by . give a detailed list of their properties. Definition The Al-Salam–Carlitz polynomials are given in terms of basic hypergeometric function In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is called h ...s by : U_n^(x;q) = (-a)^nq^_2\phi_1(q^, x^;0;q,qx/a) : V_n^(x;q) = (-a)^nq^_2\phi_0(q^, x;-;q,q^n/a) References * * Further reading * Wang, M. (2009). q-integral representation of the Al-Salam–Carlitz polynomials. Applied Mathematics Letters, 22(6), 943-945. * Askey, R., & Suslov, S. K. (1993). The q-harmonic oscillator and the Al-Salam and Carlitz polynomials. Letters in Mathematical Physics, 29(2), ...
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