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Year 2024, , 111 - 116, 28.03.2024

Sibel Koparal

https://doi.org/10.17776/csj.1365360

Abstract

References

  • [1] Benjamin A.T., Gaebler D., Gaebler R., A Combinatorial Approach to Hyperharmonic Numbers, Integers, 3 (2013) 1-9.
  • [1] Benjamin A.T., Gaebler D., Gaebler R., A Combinatorial Approach to Hyperharmonic Numbers, Integers, 3 (2013) 1-9.
  • [2] Benjamin A.T., Preston G.O., Quinn J.J., A Stirling Encounter with Harmonic Numbers, Math. Mag., 75 (2002) 95-103.
  • [2] Benjamin A.T., Preston G.O., Quinn J.J., A Stirling Encounter with Harmonic Numbers, Math. Mag., 75 (2002) 95-103.
  • [3] Caralambides C.A., Enumarative combinatorics, Chapman&Hall/Crc, Press Company, 1st ed. New York, (2002), 1-632.
  • [3] Caralambides C.A., Enumarative combinatorics, Chapman&Hall/Crc, Press Company, 1st ed. New York, (2002), 1-632.
  • [4] Cheon G.S., El-Mikkawy M., Generalized Harmonic Numbers with Riordan Arrays, J. Number Theory, 128(2) (2008) 413-425.
  • [4] Cheon G.S., El-Mikkawy M., Generalized Harmonic Numbers with Riordan Arrays, J. Number Theory, 128(2) (2008) 413-425.
  • [5] Dattoli G., Licciardi S., Sabia E., Srivastava H.M., Some Properties and Generating Functions of Generalized Harmonic Numbers, Mathematics, 7(7) (2019), Article ID 577.
  • [5] Dattoli G., Licciardi S., Sabia E., Srivastava H.M., Some Properties and Generating Functions of Generalized Harmonic Numbers, Mathematics, 7(7) (2019), Article ID 577.
  • [6] Dattoli G., Srivastava H.M., A Note on Harmonic Numbers, Umbral Calculus and Generating Functions, Appl. Math. Lett., 21 (7) (2008) 686-693.
  • [6] Dattoli G., Srivastava H.M., A Note on Harmonic Numbers, Umbral Calculus and Generating Functions, Appl. Math. Lett., 21 (7) (2008) 686-693.
  • [7] Duran Ö., Ömür N., Koparal S., On Sums with Generalized Harmonic, Hyperharmonic and Special Numbers, Miskolc Math. Notes, 21(2) (2020) 791-160.
  • [7] Duran Ö., Ömür N., Koparal S., On Sums with Generalized Harmonic, Hyperharmonic and Special Numbers, Miskolc Math. Notes, 21(2) (2020) 791-160.
  • [8] Gen ̌cev M., Binomial Sums Involving Harmonic Numbers, Math. Slovaca, 61(2) (2011) 215-226.
  • [8] Gen ̌cev M., Binomial Sums Involving Harmonic Numbers, Math. Slovaca, 61(2) (2011) 215-226.
  • [9] Koparal S., Ömür N., Südemen K.N., Some Identities for Derangement Numbers, Miskolc Math. Notes, 23(2) (2022) 773-785.
  • [9] Koparal S., Ömür N., Südemen K.N., Some Identities for Derangement Numbers, Miskolc Math. Notes, 23(2) (2022) 773-785.
  • [10] Koparal S., Ömür N., Duran Ö., On Identities Involving Generalized Harmonic, Hyperharmonic and Special Numbers with Riordan Arrays, Spec. Matrices, 9 (2021) 22-30.
  • [10] Koparal S., Ömür N., Duran Ö., On Identities Involving Generalized Harmonic, Hyperharmonic and Special Numbers with Riordan Arrays, Spec. Matrices, 9 (2021) 22-30.
  • [11] Kwon H.I., Jang G.W., Kim T., Some Identities of Derangements Numbers Arising from Differential Equations, Adv. Stud. Contemp. Math., 28(1) (2018) 73-82.
  • [11] Kwon H.I., Jang G.W., Kim T., Some Identities of Derangements Numbers Arising from Differential Equations, Adv. Stud. Contemp. Math., 28(1) (2018) 73-82.
  • [12] Ömür N., Bilgin G., Some Applications of Generalized Hyperharmonic Numbers of Order r, H_n^r (α), Adv. Appl. Math. Sci., 17(9) (2018) 617-627.
  • [12] Ömür N., Bilgin G., Some Applications of Generalized Hyperharmonic Numbers of Order r, H_n^r (α), Adv. Appl. Math. Sci., 17(9) (2018) 617-627.
  • [13] Ömür N., Koparal S., On the Matrices with the Generalized Hyperharmonic Numbers of Order r, Asian–European J. Math., 11(3) (2018) Article ID 1850045.
  • [13] Ömür N., Koparal S., On the Matrices with the Generalized Hyperharmonic Numbers of Order r, Asian–European J. Math., 11(3) (2018) Article ID 1850045.
  • [14] Ömür N., Südemen K.N., Koparal S., Some Identities with Special Numbers, Cumhuriyet Sci. J., 43(4) (2022) 696-702.
  • [14] Ömür N., Südemen K.N., Koparal S., Some Identities with Special Numbers, Cumhuriyet Sci. J., 43(4) (2022) 696-702.
  • [15] Ömür N., Koparal S., Sums Involving Generalized Harmonic and Daehee Numbers, Notes on Number Theory and Discrete Math., 28(1) (2022) 92-99.
  • [15] Ömür N., Koparal S., Sums Involving Generalized Harmonic and Daehee Numbers, Notes on Number Theory and Discrete Math., 28(1) (2022) 92-99.
  • [16] Qi F., Zhao J.L., Guo B.N., Closed Forms for Derangement Numbers in terms of the Hessenberg Determinants, Rev. R. Acad. Cienc. Exactas Fı ́s. Nat. Ser. A Mat. RACSAM, 112 (2018) 933–944.
  • [16] Qi F., Zhao J.L., Guo B.N., Closed Forms for Derangement Numbers in terms of the Hessenberg Determinants, Rev. R. Acad. Cienc. Exactas Fı ́s. Nat. Ser. A Mat. RACSAM, 112 (2018) 933–944.
  • [17] Qi F., Guo B.N., Explicit Formulas for Derangement Numbers and Their Generating Function, J. Nonlinear Funct. Anal., 2016 (2016) Article ID 45.
  • [17] Qi F., Guo B.N., Explicit Formulas for Derangement Numbers and Their Generating Function, J. Nonlinear Funct. Anal., 2016 (2016) Article ID 45.
  • [18] Rim S.H., Kim T., Pyo S.S., Identities Between Harmonic, Hyperharmonic and Daehee Numbers, J. Inequal. Appl., 2018 (2018) Article ID 168.Santmyer J.M., A Stirling like Sequence of Rational Numbers, Discrete Math., 171(1-3) (1997) 229-235.
  • [18] Rim S.H., Kim T., Pyo S.S., Identities Between Harmonic, Hyperharmonic and Daehee Numbers, J. Inequal. Appl., 2018 (2018) Article ID 168.Santmyer J.M., A Stirling like Sequence of Rational Numbers, Discrete Math., 171(1-3) (1997) 229-235.
  • [19] Sofo A., Srivastava H.M., Identities for the Harmonic Numbers and Binomial Coefficients, Ramanujan J., 25(1) (2011) 93-113.
  • [19] Sofo A., Srivastava H.M., Identities for the Harmonic Numbers and Binomial Coefficients, Ramanujan J., 25(1) (2011) 93-113.
  • [20]Şimşek Y., Special Numbers on Analytic Functions, Appl. Math., 5(7) (2014) 1091-1098.
  • [20]Şimşek Y., Special Numbers on Analytic Functions, Appl. Math., 5(7) (2014) 1091-1098.
  • [21] Wang C., Miska P., Mezö I., The r-Derangement Numbers, Discrete Math., 340(2017) 1681-1692.
  • [21] Wang C., Miska P., Mezö I., The r-Derangement Numbers, Discrete Math., 340(2017) 1681-1692.
  • [22] Choi J., Srivastava H.M., Some summation formulas involving harmonic numbers and generalized harmonic numbers, Math. Comput. Model., 54(9-10) (2011) 2220–2234.
  • [22] Choi J., Srivastava H.M., Some summation formulas involving harmonic numbers and generalized harmonic numbers, Math. Comput. Model., 54(9-10) (2011) 2220–2234.
  • [23]Simsek Y., Some classes of finite sums related to the generalized Harmonic functions and special numbers and polynomials, Montes Taurus J. Pure Appl. Math., 4(3) (2022) 61-79.
  • [23]Simsek Y., Some classes of finite sums related to the generalized Harmonic functions and special numbers and polynomials, Montes Taurus J. Pure Appl. Math., 4(3) (2022) 61-79.
  • [24]Simsek Y., New integral formulas and identities involving special numbers and functions derived from certain class of special combinatorial sums, RACSAM, 115(66) (2021) 1-14. Simsek Y., Apostol type Daehee numbers and polynomials, Adv. Stud. Contemp. Math., 26(3) (2016) 555-566.
  • [24]Simsek Y., New integral formulas and identities involving special numbers and functions derived from certain class of special combinatorial sums, RACSAM, 115(66) (2021) 1-14. Simsek Y., Apostol type Daehee numbers and polynomials, Adv. Stud. Contemp. Math., 26(3) (2016) 555-566.
  • [25]Rassias T.M., Srivastava H.M., Some classes of infinite series associated with the Riemann Zeta and Polygamma functions and generalized harmonic numbers, Appl. Math. Comput., 131(2002) 593-605.
  • [25]Rassias T.M., Srivastava H.M., Some classes of infinite series associated with the Riemann Zeta and Polygamma functions and generalized harmonic numbers, Appl. Math. Comput., 131(2002) 593-605.
  • [26]Kim T., Kim D.S., Combinatorial identities involving degenerate harmonic and hyperharmonic numbers, Adv. Appl. Math., 148(2023) Article ID 102535, 15 p.
  • [26]Kim T., Kim D.S., Combinatorial identities involving degenerate harmonic and hyperharmonic numbers, Adv. Appl. Math., 148(2023) Article ID 102535, 15 p.
  • [27] Kim T., Kim D.S., Some identities on degenerate hyperharmonic numbers, Georgian Math. J., 30(2) (2023) 255-262.
  • [27] Kim T., Kim D.S., Some identities on degenerate hyperharmonic numbers, Georgian Math. J., 30(2) (2023) 255-262.
  • [28]Dolgy D.V., Kim D.S., Kim H.K., Kim, T., Degenerate harmonic and hyperharmonic numbers, Proc. Jangjeon Math. Soc., 26(3) (2023) 259-268.
  • [28]Dolgy D.V., Kim D.S., Kim H.K., Kim, T., Degenerate harmonic and hyperharmonic numbers, Proc. Jangjeon Math. Soc., 26(3) (2023) 259-268.
  • [29]Kim D.S., Kim T., Normal ordering associated with λ-Whitney numbers of the first kind in λ-shift algebra, Russ. J. Math. Phys., 30(3) (2023) 310-319.
  • [29]Kim D.S., Kim T., Normal ordering associated with λ-Whitney numbers of the first kind in λ-shift algebra, Russ. J. Math. Phys., 30(3) (2023) 310-319.
  • [30]Kim T., Kim D.S., Some identities on degenerate r-Stirling numbers via boson operators, Russ. J. Math. Phys., 29(4) (2022) 508-517.
  • [30]Kim T., Kim D.S., Some identities on degenerate r-Stirling numbers via boson operators, Russ. J. Math. Phys., 29(4) (2022) 508-517.
  • [31] Kim T.K., Kim D.S., Some identities involving degenerate Stirling numbers associated with several degenerate polynomials and numbers, Russ. J. Math. Phys., 30(1) (2023) 62-75.
  • [31] Kim T.K., Kim D.S., Some identities involving degenerate Stirling numbers associated with several degenerate polynomials and numbers, Russ. J. Math. Phys., 30(1) (2023) 62-75.

Some Sums Involving Generalized Harmonic and r-Derangement Numbers

Year 2024, , 111 - 116, 28.03.2024

Sibel Koparal

https://doi.org/10.17776/csj.1365360

Abstract

In this paper, we derive some sums involving generalized harmonic and r-derangement numbers by using generating functions of these numbers and some combinatorial identities. The relationship between Daehee numbers and generalized harmonic numbers of rank r, H(n,r,α) is given. In addition, sums including Daehee numbers of order r,D_n^r, generalized hyperharmonic numbers of order r, H_n^r (α), Cauchy numbers of order r, C_n^r and the stirling numbers of the first kind, s(n,i) are also calculated.

Keywords

Generalized harmonic numbers 𝑟− Derangement numbers Generating functions Special numbers

References

  • [1] Benjamin A.T., Gaebler D., Gaebler R., A Combinatorial Approach to Hyperharmonic Numbers, Integers, 3 (2013) 1-9.
  • [1] Benjamin A.T., Gaebler D., Gaebler R., A Combinatorial Approach to Hyperharmonic Numbers, Integers, 3 (2013) 1-9.
  • [2] Benjamin A.T., Preston G.O., Quinn J.J., A Stirling Encounter with Harmonic Numbers, Math. Mag., 75 (2002) 95-103.
  • [2] Benjamin A.T., Preston G.O., Quinn J.J., A Stirling Encounter with Harmonic Numbers, Math. Mag., 75 (2002) 95-103.
  • [3] Caralambides C.A., Enumarative combinatorics, Chapman&Hall/Crc, Press Company, 1st ed. New York, (2002), 1-632.
  • [3] Caralambides C.A., Enumarative combinatorics, Chapman&Hall/Crc, Press Company, 1st ed. New York, (2002), 1-632.
  • [4] Cheon G.S., El-Mikkawy M., Generalized Harmonic Numbers with Riordan Arrays, J. Number Theory, 128(2) (2008) 413-425.
  • [4] Cheon G.S., El-Mikkawy M., Generalized Harmonic Numbers with Riordan Arrays, J. Number Theory, 128(2) (2008) 413-425.
  • [5] Dattoli G., Licciardi S., Sabia E., Srivastava H.M., Some Properties and Generating Functions of Generalized Harmonic Numbers, Mathematics, 7(7) (2019), Article ID 577.
  • [5] Dattoli G., Licciardi S., Sabia E., Srivastava H.M., Some Properties and Generating Functions of Generalized Harmonic Numbers, Mathematics, 7(7) (2019), Article ID 577.
  • [6] Dattoli G., Srivastava H.M., A Note on Harmonic Numbers, Umbral Calculus and Generating Functions, Appl. Math. Lett., 21 (7) (2008) 686-693.
  • [6] Dattoli G., Srivastava H.M., A Note on Harmonic Numbers, Umbral Calculus and Generating Functions, Appl. Math. Lett., 21 (7) (2008) 686-693.
  • [7] Duran Ö., Ömür N., Koparal S., On Sums with Generalized Harmonic, Hyperharmonic and Special Numbers, Miskolc Math. Notes, 21(2) (2020) 791-160.
  • [7] Duran Ö., Ömür N., Koparal S., On Sums with Generalized Harmonic, Hyperharmonic and Special Numbers, Miskolc Math. Notes, 21(2) (2020) 791-160.
  • [8] Gen ̌cev M., Binomial Sums Involving Harmonic Numbers, Math. Slovaca, 61(2) (2011) 215-226.
  • [8] Gen ̌cev M., Binomial Sums Involving Harmonic Numbers, Math. Slovaca, 61(2) (2011) 215-226.
  • [9] Koparal S., Ömür N., Südemen K.N., Some Identities for Derangement Numbers, Miskolc Math. Notes, 23(2) (2022) 773-785.
  • [9] Koparal S., Ömür N., Südemen K.N., Some Identities for Derangement Numbers, Miskolc Math. Notes, 23(2) (2022) 773-785.
  • [10] Koparal S., Ömür N., Duran Ö., On Identities Involving Generalized Harmonic, Hyperharmonic and Special Numbers with Riordan Arrays, Spec. Matrices, 9 (2021) 22-30.
  • [10] Koparal S., Ömür N., Duran Ö., On Identities Involving Generalized Harmonic, Hyperharmonic and Special Numbers with Riordan Arrays, Spec. Matrices, 9 (2021) 22-30.
  • [11] Kwon H.I., Jang G.W., Kim T., Some Identities of Derangements Numbers Arising from Differential Equations, Adv. Stud. Contemp. Math., 28(1) (2018) 73-82.
  • [11] Kwon H.I., Jang G.W., Kim T., Some Identities of Derangements Numbers Arising from Differential Equations, Adv. Stud. Contemp. Math., 28(1) (2018) 73-82.
  • [12] Ömür N., Bilgin G., Some Applications of Generalized Hyperharmonic Numbers of Order r, H_n^r (α), Adv. Appl. Math. Sci., 17(9) (2018) 617-627.
  • [12] Ömür N., Bilgin G., Some Applications of Generalized Hyperharmonic Numbers of Order r, H_n^r (α), Adv. Appl. Math. Sci., 17(9) (2018) 617-627.
  • [13] Ömür N., Koparal S., On the Matrices with the Generalized Hyperharmonic Numbers of Order r, Asian–European J. Math., 11(3) (2018) Article ID 1850045.
  • [13] Ömür N., Koparal S., On the Matrices with the Generalized Hyperharmonic Numbers of Order r, Asian–European J. Math., 11(3) (2018) Article ID 1850045.
  • [14] Ömür N., Südemen K.N., Koparal S., Some Identities with Special Numbers, Cumhuriyet Sci. J., 43(4) (2022) 696-702.
  • [14] Ömür N., Südemen K.N., Koparal S., Some Identities with Special Numbers, Cumhuriyet Sci. J., 43(4) (2022) 696-702.
  • [15] Ömür N., Koparal S., Sums Involving Generalized Harmonic and Daehee Numbers, Notes on Number Theory and Discrete Math., 28(1) (2022) 92-99.
  • [15] Ömür N., Koparal S., Sums Involving Generalized Harmonic and Daehee Numbers, Notes on Number Theory and Discrete Math., 28(1) (2022) 92-99.
  • [16] Qi F., Zhao J.L., Guo B.N., Closed Forms for Derangement Numbers in terms of the Hessenberg Determinants, Rev. R. Acad. Cienc. Exactas Fı ́s. Nat. Ser. A Mat. RACSAM, 112 (2018) 933–944.
  • [16] Qi F., Zhao J.L., Guo B.N., Closed Forms for Derangement Numbers in terms of the Hessenberg Determinants, Rev. R. Acad. Cienc. Exactas Fı ́s. Nat. Ser. A Mat. RACSAM, 112 (2018) 933–944.
  • [17] Qi F., Guo B.N., Explicit Formulas for Derangement Numbers and Their Generating Function, J. Nonlinear Funct. Anal., 2016 (2016) Article ID 45.
  • [17] Qi F., Guo B.N., Explicit Formulas for Derangement Numbers and Their Generating Function, J. Nonlinear Funct. Anal., 2016 (2016) Article ID 45.
  • [18] Rim S.H., Kim T., Pyo S.S., Identities Between Harmonic, Hyperharmonic and Daehee Numbers, J. Inequal. Appl., 2018 (2018) Article ID 168.Santmyer J.M., A Stirling like Sequence of Rational Numbers, Discrete Math., 171(1-3) (1997) 229-235.
  • [18] Rim S.H., Kim T., Pyo S.S., Identities Between Harmonic, Hyperharmonic and Daehee Numbers, J. Inequal. Appl., 2018 (2018) Article ID 168.Santmyer J.M., A Stirling like Sequence of Rational Numbers, Discrete Math., 171(1-3) (1997) 229-235.
  • [19] Sofo A., Srivastava H.M., Identities for the Harmonic Numbers and Binomial Coefficients, Ramanujan J., 25(1) (2011) 93-113.
  • [19] Sofo A., Srivastava H.M., Identities for the Harmonic Numbers and Binomial Coefficients, Ramanujan J., 25(1) (2011) 93-113.
  • [20]Şimşek Y., Special Numbers on Analytic Functions, Appl. Math., 5(7) (2014) 1091-1098.
  • [20]Şimşek Y., Special Numbers on Analytic Functions, Appl. Math., 5(7) (2014) 1091-1098.
  • [21] Wang C., Miska P., Mezö I., The r-Derangement Numbers, Discrete Math., 340(2017) 1681-1692.
  • [21] Wang C., Miska P., Mezö I., The r-Derangement Numbers, Discrete Math., 340(2017) 1681-1692.
  • [22] Choi J., Srivastava H.M., Some summation formulas involving harmonic numbers and generalized harmonic numbers, Math. Comput. Model., 54(9-10) (2011) 2220–2234.
  • [22] Choi J., Srivastava H.M., Some summation formulas involving harmonic numbers and generalized harmonic numbers, Math. Comput. Model., 54(9-10) (2011) 2220–2234.
  • [23]Simsek Y., Some classes of finite sums related to the generalized Harmonic functions and special numbers and polynomials, Montes Taurus J. Pure Appl. Math., 4(3) (2022) 61-79.
  • [23]Simsek Y., Some classes of finite sums related to the generalized Harmonic functions and special numbers and polynomials, Montes Taurus J. Pure Appl. Math., 4(3) (2022) 61-79.
  • [24]Simsek Y., New integral formulas and identities involving special numbers and functions derived from certain class of special combinatorial sums, RACSAM, 115(66) (2021) 1-14. Simsek Y., Apostol type Daehee numbers and polynomials, Adv. Stud. Contemp. Math., 26(3) (2016) 555-566.
  • [24]Simsek Y., New integral formulas and identities involving special numbers and functions derived from certain class of special combinatorial sums, RACSAM, 115(66) (2021) 1-14. Simsek Y., Apostol type Daehee numbers and polynomials, Adv. Stud. Contemp. Math., 26(3) (2016) 555-566.
  • [25]Rassias T.M., Srivastava H.M., Some classes of infinite series associated with the Riemann Zeta and Polygamma functions and generalized harmonic numbers, Appl. Math. Comput., 131(2002) 593-605.
  • [25]Rassias T.M., Srivastava H.M., Some classes of infinite series associated with the Riemann Zeta and Polygamma functions and generalized harmonic numbers, Appl. Math. Comput., 131(2002) 593-605.
  • [26]Kim T., Kim D.S., Combinatorial identities involving degenerate harmonic and hyperharmonic numbers, Adv. Appl. Math., 148(2023) Article ID 102535, 15 p.
  • [26]Kim T., Kim D.S., Combinatorial identities involving degenerate harmonic and hyperharmonic numbers, Adv. Appl. Math., 148(2023) Article ID 102535, 15 p.
  • [27] Kim T., Kim D.S., Some identities on degenerate hyperharmonic numbers, Georgian Math. J., 30(2) (2023) 255-262.
  • [27] Kim T., Kim D.S., Some identities on degenerate hyperharmonic numbers, Georgian Math. J., 30(2) (2023) 255-262.
  • [28]Dolgy D.V., Kim D.S., Kim H.K., Kim, T., Degenerate harmonic and hyperharmonic numbers, Proc. Jangjeon Math. Soc., 26(3) (2023) 259-268.
  • [28]Dolgy D.V., Kim D.S., Kim H.K., Kim, T., Degenerate harmonic and hyperharmonic numbers, Proc. Jangjeon Math. Soc., 26(3) (2023) 259-268.
  • [29]Kim D.S., Kim T., Normal ordering associated with λ-Whitney numbers of the first kind in λ-shift algebra, Russ. J. Math. Phys., 30(3) (2023) 310-319.
  • [29]Kim D.S., Kim T., Normal ordering associated with λ-Whitney numbers of the first kind in λ-shift algebra, Russ. J. Math. Phys., 30(3) (2023) 310-319.
  • [30]Kim T., Kim D.S., Some identities on degenerate r-Stirling numbers via boson operators, Russ. J. Math. Phys., 29(4) (2022) 508-517.
  • [30]Kim T., Kim D.S., Some identities on degenerate r-Stirling numbers via boson operators, Russ. J. Math. Phys., 29(4) (2022) 508-517.
  • [31] Kim T.K., Kim D.S., Some identities involving degenerate Stirling numbers associated with several degenerate polynomials and numbers, Russ. J. Math. Phys., 30(1) (2023) 62-75.
  • [31] Kim T.K., Kim D.S., Some identities involving degenerate Stirling numbers associated with several degenerate polynomials and numbers, Russ. J. Math. Phys., 30(1) (2023) 62-75.
There are 62 citations in total.

Details

Primary Language English
Subjects Numerical and Computational Mathematics (Other)
Journal Section Natural Sciences
Authors

Sibel Koparal 0000-0001-9574-9652

Publication Date March 28, 2024
Submission Date September 23, 2023
Acceptance Date November 28, 2023
Published in Issue Year 2024

Cite

APA Koparal, S. (2024). Some Sums Involving Generalized Harmonic and r-Derangement Numbers. Cumhuriyet Science Journal, 45(1), 111-116. https://doi.org/10.17776/csj.1365360