Correction: Yalcin, F.; Simsek, Y. A New Class of Symmetric Beta Type Distributions Constructed by Means of Symmetric Bernstein Type Basis Functions. Symmetry 2020, 12, 779

Fusun Yalcin; Yilmaz Simsek

doi:10.3390/sym14020282

and

Department of Mathematics, Faculty of Science University of Akdeniz, TR-07058 Antalya, Turkey

^*

Author to whom correspondence should be addressed.

Symmetry2022, 14(2), 282;https://doi.org/10.3390/sym14020282

This article belongs to the Section Mathematics

Version Notes

Order Reprints

Text Correction

There was an error in the original publication. We state that the mistakes were made in the following Section: 4. Moment Generating Function for Beta Type Distributions.

All corrections have been made in the equations of Section 4. Moment Generating Function for Beta Type Distributions, with the page involving line numbers as follows:

Page 9 line 2 from the top, page 9 line 7 from the top, page 9 line 13 from the top, page 9 Theorem 5 line 19, page 10 line 2 from the top, page 10 line 5 from the top, page 10, Theorem 6. line 10 at the top, in the proof of Theorem 6. page 10 line 5 at the bottom, in the proof of Theorem 6. page 10 line 2 at the bottom, page 11 line 4 from the top, page 11 line 8 at the top, page 11 line 5 from the bottom, page 12 in Corollary 2 line 2 at the top, page 12 in Corollary 3 line 5 at the top and page 12 in Corollary 4 line 8 at the top.

We give a corrigendum involving a mistype of not only the assertion of Theorem 5 and Theorem 6 with their proofs, but also Corollary 2, Corollary 3 and Corollary 4 of our paper.

(Ref. [1], page 9 line 2 from the top):

\begin{array}{l} M_{x} (t; a, b; n, m, k) \\ = \sum_{j = 0}^{\infty} (\sum_{c = 0}^{k} {(- 1)}^{k - c} (\begin{matrix} k \\ c \end{matrix}) \frac{B (c + j + 1, n - k + 1) - a^{k - c} b^{c + j + n - k + 1}}{{(b - a)}^{n + 1} B (n - k + 1, k + 1)}) \frac{t^{j}}{j!} . \end{array}

(Ref. [1], page 9 line 7 from the top):

M_{x} (t; a, b; n, m, k) = \sum_{j = 0}^{\infty} (\sum_{c = 0}^{k} {(- 1)}^{k - c} (\begin{matrix} k \\ c \end{matrix}) \frac{B (c + j + 1, n - k + 1) - a^{k - c} b^{c + j + n - k + 1}}{{(b - a)}^{n + 1} B (n - k + 1, k + 1)}) \frac{t^{j}}{j!} .

(Ref. [1], page 9 line 13 from the top):

\begin{array}{r} μ_{1} (a, b; n, m, k) = \sum_{c = 0}^{k} {(- 1)}^{k - c} (\begin{matrix} k \\ c \end{matrix}) a^{k - c} b^{l + c + n - k + 1} (b \\ {- a)}^{- n - 1} \frac{B (c + l + 1, n - k + 1)}{B (n - k + 1, k + 1)} \end{array}

(Ref. [1], page 9 line 19, Theorem 5):

\begin{array}{l} E (e^{- t l n (x - a)}) \\ = \sum_{v = 0}^{\infty} \sum_{j = 0}^{v} \sum_{l = 0}^{v} {(- 1)}^{v - l + j} (\begin{matrix} v \\ l \end{matrix}) \frac{{(b - a)}^{m + l} S_{1} (v, j) B (k + l + 1, n - k + 1)}{v! B (n - k + 1, k + 1)} t^{j} \end{array}

(Ref. [1], page 10 line 2 from the top):

\begin{array}{l} E (e^{- t l n (x - a)}) \\ = \sum_{j = 0}^{\infty} {(- 1)}^{j} (\begin{matrix} v \\ l \end{matrix}) t^{j} \sum_{v = 0}^{\infty} \frac{S_{1} (v, j)}{v! {(b - a)}^{n - m + 1}} \int_{a}^{b} \frac{{(x - a - 1)}^{v} {(x - a)}^{k} {(b - x)}^{n - k}}{B (n - k + 1, k + 1)} d x . \end{array}

(Ref. [1], page 10 line 5 from the top):

E (e^{- t l n (x - a)}) = \sum_{j = 0}^{\infty} \sum_{v = 0}^{\infty} \frac{{(- 1)}^{j} t^{j} S_{1} (v, j)}{v! B (n - k + 1, k + 1) {(b - a)}^{n - m + 1}} \sum_{l = 0}^{v} {(- 1)}^{v - l} (\begin{matrix} v \\ l \end{matrix})

(Ref. [1], page 10 line 10, Theorem 6 from the top):

\begin{array}{l} \sum_{d = 0}^{\infty} \sum_{k = 0}^{d} (\begin{matrix} d \\ k \end{matrix}) \frac{{(- a - 1)}^{d - k} μ_{k}}{d!} {(- t)}_{(d)} \\ = \sum_{v = 0}^{\infty} \sum_{j = 0}^{v} \sum_{l = 0}^{v} {(- 1)}^{v - l + j} (\begin{matrix} v \\ l \end{matrix}) \frac{{(b - a)}^{m + l} S_{1} (v, j) B (k + l + 1, n - k + 1)}{v! B (n - k + 1, k + 1)} t^{j} \end{array}

(Ref. [1], in the proof of Theorem 6. page 10 line 5 from the bottom):

\begin{array}{l} E (\sum_{c = 0}^{\infty} \sum_{d = 0}^{\infty} S_{1} (d, c) \frac{{(x - a - 1)}^{d}}{d!} {(- t)}^{c}) \\ = \sum_{v = 0}^{\infty} \sum_{j = 0}^{v} \sum_{l = 0}^{v} {(- 1)}^{v - l + j} (\begin{matrix} v \\ l \end{matrix}) \frac{{(b - a)}^{m + l} S_{1} (v, j) B (k + l + 1, n - k + 1)}{v! B (n - k + 1, k + 1)} t^{j} \end{array}

(Ref. [1], in the proof of Theorem 6. page 10 line 2 from the bottom):

\begin{array}{l} \sum_{d = 0}^{\infty} \sum_{c = 0}^{d} \sum_{k = 0}^{d} (\begin{matrix} d \\ k \end{matrix}) S_{1} (d, c) \frac{{(- a - 1)}^{d - k} E (x^{k})}{d!} {(- t)}^{c} \\ = \sum_{v = 0}^{\infty} \sum_{j = 0}^{v} \sum_{l = 0}^{v} {(- 1)}^{v - l + j} (\begin{matrix} v \\ l \end{matrix}) \frac{{(b - a)}^{m + l} S_{1} (v, j) B (k + l + 1, n - k + 1)}{v! B (n - k + 1, k + 1)} t^{j} \end{array}

(Ref. [1], page 11 line 4 from the top):

B (n + 1 - k, k + 1) (β_{0} P_{0} (- t) + β_{1} P_{1} (- t) + \dots + β_{d} P_{d} (- t) + \dots) = \sum_{v = 0}^{\infty} \frac{{(- 1)}^{v} {(b - a)}^{v + m - 1}}{v!} (S_{1} (v, 0) - S_{1} (v, 1) t + \dots + {(- 1)}^{v} S_{1} (v, v) t^{v}) (B (v + k + 1, n + 1 - k) + \dots + {(- 1)}^{v} \frac{B (k + 1, n + 1 - k)}{{(b - a)}^{v}},

(Ref. [1], page 11 line 8 at the top)

where

β_{d} = \sum_{k = 0}^{d} (\begin{matrix} d \\ k \end{matrix}) \frac{{(- a - 1)}^{d - k} μ_{k}}{d!}

and

P_{d} (t) = t (t - 1) \dots (t - d + 1) = \sum_{m = 0}^{d} S_{1} (d, m) t^{m} .

(Ref. [1], page 11 line 5 from the bottom):

\begin{array}{l} \sum_{d = 0}^{\infty} \sum_{k = 0}^{d} (\begin{matrix} d \\ k \end{matrix}) \frac{{(- a - 1)}^{d - k} μ_{k}}{d!} {(- t)}_{(d)} \\ = \sum_{v = 0}^{\infty} \sum_{l = 0}^{v} {(- 1)}^{v - l} (\begin{matrix} v \\ l \end{matrix}) \frac{{(b - a)}^{m + l} B (k + l + 1, n - k + 1)}{v! B (n - k + 1, k + 1)} \sum_{j = 0}^{v} S_{1} (v, j) {(- t)}^{j} \end{array}

(Ref. [1], page 12 line 2 from the top, Corollary 2):

\sum_{d = 0}^{\infty} β_{d} P_{d} (- t) = \sum_{v = 0}^{\infty} {(- t)}_{(v)} \sum_{l = 0}^{v} {(- 1)}^{v - l} (\begin{matrix} v \\ l \end{matrix}) \frac{(k + 1)! (n + 1)! {(b - a)}^{m + l}}{k! (n + l + 1)!}

(Ref. [1], page 12 line 5 from the top, Corollary 3):

\begin{array}{l} \sum_{d = 0}^{\infty} β_{d} P_{d} (- t) \\ = \sum_{v = 0}^{\infty} \sum_{j = 0}^{v} L (v, j) t_{(j)} \sum_{l = 0}^{v} {(- 1)}^{v - l} (\begin{matrix} v \\ l \end{matrix}) \frac{(k + 1)! (n + 1)! {(b - a)}^{m + l}}{k! (n + l + 1)!} \end{array}

(Ref. [1], page 12 line 8 from the top, Corollary 4):

\sum_{d = 0}^{\infty} β_{d} P_{d} (- t) = \sum_{v = 0}^{\infty} \sum_{l = 0}^{v} {(- 1)}^{v - l} (\begin{matrix} v \\ l \end{matrix}) \frac{(k + 1)! (n + 1)! {(b - a)}^{m + l}}{k! (n + l + 1)!} P_{v} (- t)

The authors apologize for any inconvenience caused and state that the scientific conclusions are unaffected. The original publication has also been updated.

Reference

Yalcin, F.; Simsek, Y. A new class of symmetric beta type distributions constructed by means of symmetric bernstein type basis functions. Symmetry 2020, 12, 779. [Google Scholar] [CrossRef]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

Correction: Yalcin, F.; Simsek, Y. A New Class of Symmetric Beta Type Distributions Constructed by Means of Symmetric Bernstein Type Basis Functions. Symmetry 2020, 12, 779

Text Correction

Reference

Article Metrics

Citations

Article Access Statistics