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By Stephen J. McPhee

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Bell, Riordan, Rota-School started in 1906 in San Francisco (the year of the big earthquake), when Eric Temple Bell (1883–1960) read some books on number theory [430, p. 109]. The first one was by Paul Bachmann (1837– 1920) who enriched the whole theory with detailed proofs. Bachmann had a doctorate from Berlin 1862; his instructors were Ernst Kummer (1810–1893) and Martin Ohm (1792–1872) (Bachmann himself had no doctoral students). The second book Bell read was Théorie des Nombres [363] by E. Lucas (1842–1891) [363]; here Bell was initiated into the so-called umbral calculus.

52) 0 < |q| < 1, a ∈ R. 53) f (t, q) dq (t), 0 where a 0 ∞ f (t, q) dq (t) ≡ a(1 − q) f aq n , q q n , n=0 24 1 Introduction The second, which is the unit of measurement for the time, may be compared to the Cigler -operator. This operator is the simplest operator in q-analysis; it is called σ in French literature and η in the papers by Rogers and Jackson and in Hardy and Wright [268, p. 386]. 1s Definition 11 On the field of polynomials, we define [ x ]f (x) ≡ f (x) ≡ f (qx). 54) Let q ∈ C , |q| == 1 and Mer(C ) be the field of meromorphic functions in C .

495]. Four features of this q-umbral-calculus are the logarithmic behavior, the formal power series, the formal calculations and the frequent use of the infinity symbol. 19) with all hypergeometric formulas. This is achieved by means of the q-factorial and the associated q-hypergeometric series. The q function is closely associated with this. Definition 1 The variables a, b, c, a1 , a2 , . . , b1 , b2 , . . ∈ C denote certain parameters in hypergeometric series or q-hypergeometric series. The variables i, j, k, l, m, n, p, r will denote natural numbers, except for certain cases where it will be clear from the context that i denotes the imaginary unit.

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