General order multivariate Padé approximants for pseudo-multivariate functions
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Bibliographic record
Abstract
Although general order multivariate Padé approximants were introduced some decades ago, very few explicit formulas for special functions have been given. We explicitly construct some general order multivariate Padé approximants to the class of so-called pseudo-multivariate functions, using the Padé approximants to their univariate versions. We also prove that the constructed approximants inherit the normality and consistency properties of their univariate relatives, which do not hold in general for multivariate Padé approximants. Examples include the multivariate forms of the exponential and the <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="q"> <mml:semantics> <mml:mi>q</mml:mi> <mml:annotation encoding="application/x-tex">q</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -exponential functions <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E left-parenthesis x comma y right-parenthesis equals sigma-summation Underscript i comma j equals 0 Overscript normal infinity Endscripts StartFraction x Superscript i Baseline y Superscript j Baseline Over left-parenthesis i plus j right-parenthesis factorial EndFraction"> <mml:semantics> <mml:mrow> <mml:mi>E</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:munderover> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:munderover> <mml:mfrac> <mml:mrow> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msup> <mml:msup> <mml:mi>y</mml:mi> <mml:mi>j</mml:mi> </mml:msup> </mml:mrow> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>i</mml:mi> <mml:mo>+</mml:mo> <mml:mi>j</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>!</mml:mo> </mml:mrow> </mml:mfrac> </mml:mrow> <mml:annotation encoding="application/x-tex">E\left ( x,y\right ) =\sum _{i,j=0}^\infty \frac {x^iy^j}{\left ( i+j\right ) !}</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> and <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper E Subscript q Baseline left-parenthesis x comma y right-parenthesis equals sigma-summation Underscript i comma j equals 0 Overscript normal infinity Endscripts StartFraction x Superscript i Baseline y Superscript j Baseline Over left-bracket i plus j right-bracket Subscript q Baseline factorial EndFraction comma"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>q</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:munderover> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:munderover> <mml:mfrac> <mml:mrow> <mml:msup> <mml:mi>x</mml:mi> <mml:mi>i</mml:mi> </mml:msup> <mml:msup> <mml:mi>y</mml:mi> <mml:mi>j</mml:mi> </mml:msup> </mml:mrow> <mml:mrow> <mml:mo stretchy="false">[</mml:mo> <mml:mi>i</mml:mi> <mml:mo>+</mml:mo> <mml:mi>j</mml:mi> <mml:msub> <mml:mo stretchy="false">]</mml:mo> <mml:mi>q</mml:mi> </mml:msub> <mml:mo>!</mml:mo> </mml:mrow> </mml:mfrac> <mml:mo>,</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">E_q\left ( x,y\right ) =\sum _{i,j=0}^\infty \frac {x^iy^j}{[i+j]_q!},</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> as well as the Appell function <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper F 1 left-parenthesis a comma 1 comma 1 semicolon c semicolon x comma y right-parenthesis equals sigma-summation Underscript i comma j equals 0 Overscript normal infinity Endscripts StartFraction left-parenthesis a right-parenthesis Subscript i plus j Baseline x Superscript i Baseline y Superscript j Baseline Over left-parenthesis c right-parenthesis Subscript i plus j Baseline EndFraction"> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi>F</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>;</mml:mo> <mml:mi>c</mml:mi> <mml:mo>;</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:munderover> <mml:mo> ∑ </mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>i</mml:mi> <mml:mo>,</mml:mo> <mml:mi>j</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:mi mathvariant="normal"> ∞ </mml:mi>
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it