| Summary: | This contribution deals with the sequence <inline-formula><math display="inline"><semantics><msub><mrow><mo>{</mo><msubsup><mi mathvariant="double-struck">U</mi><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow></msubsup><mrow><mo>(</mo><mi>x</mi><mo>;</mo><mi>q</mi><mo>,</mo><mi>j</mi><mo>)</mo></mrow><mo>}</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></semantics></math></inline-formula> of monic polynomials in <i>x</i>, orthogonal with respect to a Sobolev-type inner product related to the Al-Salam–Carlitz I orthogonal polynomials, and involving an arbitrary number <i>j</i> of <i>q</i>-derivatives on the two boundaries of the corresponding orthogonality interval, for some fixed real number <inline-formula><math display="inline"><semantics><mrow><mi>q</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></semantics></math></inline-formula>. We provide several versions of the corresponding connection formulas, ladder operators, and several versions of the second order <i>q</i>-difference equations satisfied by polynomials in this sequence. As a novel contribution to the literature, we provide certain three term recurrence formula with rational coefficients satisfied by <inline-formula><math display="inline"><semantics><mrow><msubsup><mi mathvariant="double-struck">U</mi><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>a</mi><mo>)</mo></mrow></msubsup><mrow><mo>(</mo><mi>x</mi><mo>;</mo><mi>q</mi><mo>,</mo><mi>j</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, which paves the way to establish an appealing generalization of the so-called <i>J</i>-fractions to the framework of Sobolev-type orthogonality.
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