https://nzjmath.org/index.php/NZJMATH <p>Welcome to the official website of the New Zealand Journal of Mathematics. From 2020, the old website will be gradually phased out and past issues will be uploaded to this website.</p> <p>The New Zealand Journal of Mathematics seeks to publish high quality research papers in diverse areas of pure and applied mathematics. Well-written survey articles are also warmly invited.</p> <p>Papers should be of general interest and of moderate length. The journal is more likely to publish papers on topics that overlap with the interests of the Editorial Board or that are of interest to at least one mathematician in New Zealand.</p> <p>The online ISSN is 1179-4984.</p> New Zealand Mathematical Society and Department of Mathematics at the University of Auckland en-US New Zealand Journal of Mathematics 1179-4984 https://nzjmath.org/index.php/NZJMATH/article/view/579 <p>We improve recent results of D. Gomez and A. Winterhof (2010) and of A. Ostafe and I. E. Shparlinski (2011) on the Waring problem with Dickson polynomials in the case of prime finite fields. Our approach is based on recent bounds of Kloosterman and Gauss sums due to A. Ostafe, I. E. Shparlinski and J. F. Voloch (2021).</p> Igor E. Shparlinski Jose Felipe Voloch Copyright (c) 2025 Author 2025-07-15 2025-07-15 56 1 13 10.53733/579 https://nzjmath.org/index.php/NZJMATH/article/view/436 <p>We provide elliptic extensions of elementary identities such as the sum of the first $n$ odd or even numbers, the geometric sum and the sum of the first $n$ cubes. Many such identities, and their $q$-analogues, are indefinite sums, and can be obtained from telescoping. So we used telescoping in our study to find elliptic extensions of these identities. In the course of our study, we obtained an identity with many parameters, which appears to be new even in the $q$-case. <br />In addition, we recover some $q$-identities due to Warnaar. </p> Gaurav Bhatnagar Archna Kumari Michael Schlosser Copyright (c) 2025 Author 2025-07-15 2025-07-15 56 15 32 10.53733/436 https://nzjmath.org/index.php/NZJMATH/article/view/465 <p>In this paper we express two new mock theta functions with one parameter as the Appell-Lerch sums using the Bailey mechanism. Meanwhile, we also obtain some Hecke-type double sums for some new $q$-series. In addition, we establish the relationships between the mock theta functions and the classical sixth and eighth order mock theta functions.<br />Furthermore, we give the Hecke-type double sums for the second order mock theta function $D_5(q)$.</p> Li-Jun Hao Liang-Liang Xu Copyright (c) 2025 Author 2025-07-15 2025-07-15 56 33 43 10.53733/465