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<mml:mi>x<\/mml:mi>\n                                  <mml:mrow>\n                                    <mml:mi>n<\/mml:mi>\n                                    <mml:mo>-<\/mml:mo>\n                                    <mml:mn>1<\/mml:mn>\n                                  <\/mml:mrow>\n                                <\/mml:msub>\n                                <mml:msub>\n                                  <mml:mi>x<\/mml:mi>\n                                  <mml:mrow>\n                                    <mml:mi>n<\/mml:mi>\n                                    <mml:mo>-<\/mml:mo>\n                                    <mml:mn>2<\/mml:mn>\n                                  <\/mml:mrow>\n                                <\/mml:msub>\n                                <mml:mo>-<\/mml:mo>\n                                <mml:mi>a<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mrow>\n                                <mml:msub>\n                                  <mml:mi>x<\/mml:mi>\n                                  <mml:mrow>\n                                    <mml:mi>n<\/mml:mi>\n                                    <mml:mo>-<\/mml:mo>\n                                    <mml:mn>1<\/mml:mn>\n                                  <\/mml:mrow>\n                                <\/mml:msub>\n                                <mml:mo>-<\/mml:mo>\n                                <mml:msub>\n                                  <mml:mi>x<\/mml:mi>\n                                  <mml:mrow>\n                                    <mml:mi>n<\/mml:mi>\n                                    <mml:mo>-<\/mml:mo>\n                                    <mml:mn>2<\/mml:mn>\n                                  <\/mml:mrow>\n                                <\/mml:msub>\n                              <\/mml:mrow>\n                            <\/mml:mfrac>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mspace\/>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\u2208<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>N<\/mml:mi>\n                              <mml:mn>0<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:mtd>\n                      <\/mml:mtr>\n                    <\/mml:mtable>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:disp-formula>where <jats:inline-formula><jats:alternatives><jats:tex-math>$$a\\in \\left[ 0,\\infty \\right) $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>a<\/mml:mi>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:mfenced>\n                      <mml:mn>0<\/mml:mn>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mi>\u221e<\/mml:mi>\n                    <\/mml:mfenced>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> and the initial values <jats:inline-formula><jats:alternatives><jats:tex-math>$$x_{-2}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>x<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>-<\/mml:mo>\n                      <mml:mn>2<\/mml:mn>\n                    <\/mml:mrow>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:tex-math>$$x_{-1}$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>x<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>-<\/mml:mo>\n                      <mml:mn>1<\/mml:mn>\n                    <\/mml:mrow>\n                  <\/mml:msub>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are real numbers, is solvable in closed form and that solutions can be analyzed in detail by the Fibonacci numbers.<\/jats:p>","DOI":"10.1007\/s12190-024-02085-5","type":"journal-article","created":{"date-parts":[[2024,4,25]],"date-time":"2024-04-25T13:02:36Z","timestamp":1714050156000},"page":"3053-3099","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Solvability and solution character of a hyperbolic cotangent-type difference equation of second-order"],"prefix":"10.1007","volume":"70","author":[{"given":"Durhasan 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