{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,25]],"date-time":"2025-09-25T14:33:33Z","timestamp":1758810813130,"version":"3.37.3"},"reference-count":21,"publisher":"Springer Science and Business Media LLC","issue":"1-2","license":[{"start":{"date-parts":[[2022,9,4]],"date-time":"2022-09-04T00:00:00Z","timestamp":1662249600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,9,4]],"date-time":"2022-09-04T00:00:00Z","timestamp":1662249600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Queueing Syst"],"published-print":{"date-parts":[[2022,10]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>In this paper, we study a functional equation for generating functions of the form <jats:inline-formula><jats:alternatives><jats:tex-math>$$f(z) = g(z) \\sum _{i=1}^M p_i f(\\alpha _i(z)) + K(z)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>f<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>z<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:mi>g<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>z<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:msubsup>\n                      <mml:mo>\u2211<\/mml:mo>\n                      <mml:mrow>\n                        <mml:mi>i<\/mml:mi>\n                        <mml:mo>=<\/mml:mo>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:mrow>\n                      <mml:mi>M<\/mml:mi>\n                    <\/mml:msubsup>\n                    <mml:msub>\n                      <mml:mi>p<\/mml:mi>\n                      <mml:mi>i<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mi>f<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>\u03b1<\/mml:mi>\n                        <mml:mi>i<\/mml:mi>\n                      <\/mml:msub>\n                      <mml:mrow>\n                        <mml:mo>(<\/mml:mo>\n                        <mml:mi>z<\/mml:mi>\n                        <mml:mo>)<\/mml:mo>\n                      <\/mml:mrow>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>+<\/mml:mo>\n                    <mml:mi>K<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>z<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>, viz. a recursion with multiple recursive terms. We derive and analyze the solution of this equation for the case that the <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha _i(z)$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:msub>\n                      <mml:mi>\u03b1<\/mml:mi>\n                      <mml:mi>i<\/mml:mi>\n                    <\/mml:msub>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>z<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula> are commutative contraction mappings. The results are applied to a wide range of queueing, autoregressive and branching processes.<\/jats:p>","DOI":"10.1007\/s11134-022-09861-9","type":"journal-article","created":{"date-parts":[[2022,9,4]],"date-time":"2022-09-04T11:02:23Z","timestamp":1662289343000},"page":"7-23","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Functional equations with multiple recursive terms"],"prefix":"10.1007","volume":"102","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4493-6367","authenticated-orcid":false,"given":"Ivo","family":"Adan","sequence":"first","affiliation":[]},{"given":"Onno","family":"Boxma","sequence":"additional","affiliation":[]},{"given":"Jacques","family":"Resing","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,9,4]]},"reference":[{"key":"9861_CR1","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1007\/s11134-018-9592-z","volume":"91","author":"IJBF Adan","year":"2019","unstructured":"Adan, I.J.B.F., Hathaway, B., Kulkarni, V.G.: On first-come, first-served queues with two classes of impatient customers. 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