Elementary hypergeometric functions, Heun functions, and moments of MKZ operators
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Elementary hypergeometric functions, Heun functions, and moments of MKZ operators Ana-Maria Acu1
· Ioan Rasa2
Received: 16 February 2020 / Accepted: 28 September 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract We consider some hypergeometric functions and prove that they are elementary functions. Consequently, the second order moments of Meyer-König and Zeller type operators are elementary functions. The higher order moments of these operators are expressed in terms of elementary functions and polylogarithms. Other applications are concerned with the expansion of certain Heun functions in series or finite sums of elementary hypergeometric functions. Keywords hypergeometric functions · elementary functions · Meyer-König and Zeller type operators · polylogarithms · Heun functions Mathematics Subject Classification 33C05 · 33C90 · 33E30 · 41A36
1 Introduction In 1960 Meyer-König and Zeller [23] introduced a sequence of positive linear operators defined for f ∈ C[0, 1] as follows ⎧ ∞ ⎪ k ⎨ n + k x k (1 − x)n+1 f , x ∈ [0, 1), Mn ( f ; x) = n+k k k=0 ⎪ ⎩ f (1), x = 1. This paper is devoted to some families of elementary hypergeometric functions, with applications to the moments of Meyer-König and Zeller type operators and to the expansion of certain Heun functions in series or finite sums of elementary hypergeometric functions.
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Ana-Maria Acu [email protected] Ioan Rasa [email protected]
1
Department of Mathematics and Informatics, Lucian Blaga University of Sibiu, Str. Dr. I. Ratiu, No.5-7, 550012 Sibiu, Romania
2
Faculty of Automation and Computer Science, Department of Mathematics, Technical University of Cluj-Napoca, Str. Memorandumului nr. 28, Cluj-Napoca, Romania 0123456789().: V,-vol
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A.-M. Acu , I. Rasa
In [4], J.A.H. Alkemade proved that the second order moment of the Meyer-König and Zeller operators can be expressed as Mn (e2 ; x) = x 2 +
x(1 − x)2 2 F1 (1, 2; n + 2; x), x ∈ [0, 1). n+1
(1.1)
Here er (t) := t r , t ∈ [0, 1], r ≥ 0, and 2 F1 (a, b; c; x) denotes the hypergeometric function. A closed form of Mn (e2 ; x), showing that it is an elementary function, was given in [14]. It is said in [14, p.2]: “ The opinion that the second-order moment of the celebrated MeyerKönig and Zeller operators is a non-elementary function was tacitly accepted since 1960.” The same opinion was firmly expressed in [13, Sect. 2]. This is surprising, because it is elementary to prove that 2 F1 (1, 2; n + 2; x) is elementary: such proofs will be given in the next sections. Alternatively, we can use 2 F1 (1, 2; 3; x)
= −2x −2 (x + log(1 − x)),
(1.2)
and [12, 2.8(24)]: 2 F1 (a, b; c
+ m; x) =
(m) (c)m (1 − x)m+c−a−b (1 − x)a+b−c 2 F1 (a, b; c; x) , (1.3) (c − a)m (c − b)m
where (r )m := r (r + 1) . . . (r + m − 1), m ≥ 1, and (r )0 := 1. They lead to 2 F1 (1, 2; n
(3)n−1 (1 − x)n−1 (2 F1 (1, 2; 3; x))(n−1) (2)n−1 (1)n−1 (n−1) n+1 =− , (1 − x)n−1 x −1 + x −2 log(1 − x) (n − 1)!
+ 2; x) =
(1.4)
which is obviously an elementary function. Thus Mn (e2 ; x)
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