Erratum to: On the Characterisations of a New Class of Strong Uniqueness Polynomials Generating Unique Range Sets
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Erratum to: On the Characterisations of a New Class of Strong Uniqueness Polynomials Generating Unique Range Sets Abhijit Banerjee1 · Sanjay Mallick1
© Springer-Verlag GmbH Germany 2017
Erratum to: Comput. Methods Funct. Theory (2017) 17:19–45 DOI 10.1007/s40315-016-0174-y In the proof of Lemma 2.6 Case 2 has to be replaced as follows.
Case 2 Let A = ωl for some l such that 0 ≤ l ≤ m − 1. Then also F(t0 ) = 0 = F (t0 ) implies ent0 = A and emt0 = 1. Now, if possible, suppose that there exist more than one t0 such that emt0 = 1 and ent0 = A, i.e., there exist t0 p , t0q with et0 p = et0q such that emt0 p = 1 = emt0q and ent0 p = A = ent0q , i.e., em(t0 p −t0q ) = 1 and en(t0 p −t0q ) = 1, i.e., m(t0 p − t0q ) = 2k1 πi for some k1 ∈ Z and n(t0 p − t0q ) = 2k2 πi for some k2 ∈ Z. Since gcd(m, n) = 1, so there exists x, y ∈ Z such that mx + ny = 1, i.e., m(t0 p − t0q )x + n(t0 p − t0q )y = (t0 p − t0q ), i.e., 2k1 πi x + 2k2 πi y = (t0 p − t0q ), i.e., 2πi(xk1 + yk2 ) = (t0 p − t0q ), i.e., 2sπi = (t0 p − t0q ), where s = xk1 + yk2 ∈ Z. Therefore et0 p = etoq , which is a contradiction to et0 p = et0q . Therefore φ(et ), hence φ(z), has exactly one multiple zero ω j , where 0 ≤ j ≤ m − 1 and ωm j = 1, ωn j = ωl and that is of multiplicity 4. Now in particular if A = 1, then we have ω j is the multiple zero of φ(z) for some j ∈ {0, 1, . . . , m − 1} such that ωm j = 1 and ωn j = 1 i.e., ω j = 1 as gcd(m, n) = 1.
Communicated by Risto Korhonen. The online version of the original article can be found under doi:10.1007/s40315-016-0174-y.
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Abhijit Banerjee [email protected]; [email protected] Sanjay Mallick [email protected]
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Department of Mathematics, University of Kalyani, Kalyani, West Bengal 741235, India
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